# Canonical Sets of BestL1-Approximation

Authors: Dimiter Dryanov; Petar Petrov
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

In mathematics, the termapproximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.


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## Body

## 1. Canonical Interpolation Sets of BestL1-Approximation

We start with the notion of a canonical set of bestL1-approximation. Let μ be a positive Borel measure defined on the compact set K⊂Rd such that the quantity ∥f∥1:=∫K|f|dμ represents a norm in the linear space C(K) of functions that are continuous on K. Let U           be a linear subspace of C(K).


### 1.1. The Problem of BestL1(μ)-Approximation

Given a functionf∈C(K), then we have the following.(A)
BestL1(μ)-approximation of f by elements of U: find a function uf∈U such that
(1.1)∥f-uf∥1≤∥f-u∥1∀u∈U.(B)
Best one-sided from aboveL1(μ)-approximation by elements of U: find a function uf*∈U,   uf*≥f on K, such that
(1.2)∥f-uf*∥1≤∥f-u∥1∀u∈U,u≥fonK.(C)
Best one-sided from belowL1(μ)-approximation by elements of U: find a function u*f∈U,   u*f≤f on K, such that
(1.3)∥f-u*f∥1≤∥f-u∥1∀u∈U,u≤fonK.
Any solutionuf of the approximation problem (A) is called a best L1(μ)-approximant to f from U. Any solution of the approximation problem (B) is called a best one-sided from above L1(μ)-approximant to f from U; respectively from below for the approximation problem (C). When μ is the usual Lebesgue measure, we use the notation L1-approximation, respectively,  L1-approximant.
For qualitative results on bestL1(μ)-approximation from finite-dimensional subspaces see [1].


### 1.2. Lagrange Interpolation Problem

Given a functionf∈C(K). We call a subset X⊂Kunisolvent forU if the interpolation problem
(1.4)u(x)=f(x),x∈X
possesses a unique solution u∈U for every f∈C(K).


### 1.3. Canonical Sets of BestL1-Approximation

Let𝒞⊂C(K) be a class of functions. We say that a set X⊂K, which is unisolvent for U, is a canonical set of bestL1(μ)-approximation to 𝒞, if for all f∈𝒞 the solution of the interpolation problem (1.4) is a best L1(μ)-approximant to f from U. Analogously, a unisolvent set X*⊂K for U (resp., X*⊂K) is called a canonical set of best one-sided from above(resp., from below) L1(μ)-approximation to 𝒞, if for all f∈𝒞 the solution of the interpolation problem (1.4) with X replaced by X* (resp., X*) is a best one-sided from above (resp., from below)  L1(μ)-approximant to f from U. In the case of one-sided L1(μ)-approximation the interpolation problem (1.4) is considered in a broader Lagrange-Hermite interpolation sense: for all points x∈X which are interior for K we require matching not only the values of u and f but also the values of their first (partial) derivatives.


## 1.1. The Problem of BestL1(μ)-Approximation

Given a functionf∈C(K), then we have the following.(A)
BestL1(μ)-approximation of f by elements of U: find a function uf∈U such that
(1.1)∥f-uf∥1≤∥f-u∥1∀u∈U.(B)
Best one-sided from aboveL1(μ)-approximation by elements of U: find a function uf*∈U,   uf*≥f on K, such that
(1.2)∥f-uf*∥1≤∥f-u∥1∀u∈U,u≥fonK.(C)
Best one-sided from belowL1(μ)-approximation by elements of U: find a function u*f∈U,   u*f≤f on K, such that
(1.3)∥f-u*f∥1≤∥f-u∥1∀u∈U,u≤fonK.
Any solutionuf of the approximation problem (A) is called a best L1(μ)-approximant to f from U. Any solution of the approximation problem (B) is called a best one-sided from above L1(μ)-approximant to f from U; respectively from below for the approximation problem (C). When μ is the usual Lebesgue measure, we use the notation L1-approximation, respectively,  L1-approximant.
For qualitative results on bestL1(μ)-approximation from finite-dimensional subspaces see [1].


## 1.2. Lagrange Interpolation Problem

Given a functionf∈C(K). We call a subset X⊂Kunisolvent forU if the interpolation problem
(1.4)u(x)=f(x),x∈X
possesses a unique solution u∈U for every f∈C(K).


## 1.3. Canonical Sets of BestL1-Approximation

Let𝒞⊂C(K) be a class of functions. We say that a set X⊂K, which is unisolvent for U, is a canonical set of bestL1(μ)-approximation to 𝒞, if for all f∈𝒞 the solution of the interpolation problem (1.4) is a best L1(μ)-approximant to f from U. Analogously, a unisolvent set X*⊂K for U (resp., X*⊂K) is called a canonical set of best one-sided from above(resp., from below) L1(μ)-approximation to 𝒞, if for all f∈𝒞 the solution of the interpolation problem (1.4) with X replaced by X* (resp., X*) is a best one-sided from above (resp., from below)  L1(μ)-approximant to f from U. In the case of one-sided L1(μ)-approximation the interpolation problem (1.4) is considered in a broader Lagrange-Hermite interpolation sense: for all points x∈X which are interior for K we require matching not only the values of u and f but also the values of their first (partial) derivatives.


## 2. Canonical Sets of Univariate BestL1-Approximation

For details on the results in the present section see [1–6]. In the univariate case, when the set K is an interval [a,b] and the subspace U is finite-dimensional, the problem of existence and characterization of a canonical set is well studied and is closely related to the notion of a Haar-Chebyshev system (see [1] for details).


### 2.1. Haar-Chebyshev System of Ordern

A set ofn functions {u1,u2,…,un}⊂C[a,b] is a Haar-Chebyshev system (T-system) of order n on the interval [a,b] if each nontrivial linear combination of u1,u2,…,un has at most n-1 zeros in  [a,b].  In other words, a set of n functions {u1,u2,…,un} is a Haar-Chebyshev system of order n on the interval [a,b] if it is linearly independent with respect to an arbitrary chosen set of n distinct points in [a,b]. We say that an n-dimensional subspace U of  C[a,b] is a Haar-Chebyshev space(T-space)of ordern on [a,b] if every  u∈U, u≢0, has no more than n-1 distinct zeros in [a,b]. Equivalently, U is a T-space of order n on [a,b], if for every basis u1,u2,…,un of U there exists an ε∈{-1,1} such that εdet{ui(xj)}i,j=1n>0  for all  a≤x1<x2<⋯<xn≤b. It is also often said that the basis functions u1,u2,…,un constitute a T-space of order n on [a,b].


### 2.2. Characterization of BestL1(μ)-Approximation by Canonical Interpolation Sets

Following [1], throughout this section we suppose that μ is a finite positive nonatomic measure. The problem of best L1(μ)-approximation by a Haar-Chebyshev space has an elegant solution via interpolation on a canonical set. Let U⊂C[a,b] be a Haar-Chebyshev space of order n and let u1,u2,…,un be a basis of U. Consider the convex cone 𝒞+(U) defined as
(2.1)𝒞+(U):={un+1:det{ui(xj)}i,j=1n+1≥0}
for all a≤x1<x2<⋯<xn<xn+1≤b.Theorem 2.1.
LetU⊂C[a,b] be a Haar-Chebyshev space of order n on [a,b]. Then the following holds true.(a)
Uniqueness of the Canonical Set. There is a unique set of pointsa=x0<x1<⋯<xn+1=b such that
(2.2)∑j=1n+1(-1)j∫xj-1xjudμ=0,∀u∈U.(b)
BestL1(μ)-Approximation via Interpolation. Let f∈𝒞+(U). Then the unique solution uf∈U of the interpolation problem
(2.3)uf(xj)=f(xj),j=1,2,…,n
is the unique best L1(μ)-approximant to f from U.Remark 2.2.
Following Theorem2.1, the set {xk}k=1n is a canonical set of best L1(μ)-approximation to the convex cone 𝒞+(U) of functions from the T-space U. In many cases the approximating subspace U can be a kernel of a certain differential operator 𝔇, that is, U=Ker𝔇 and the convex cone 𝒞𝔇+(U) of functions to be approximated can be defined by
(2.4)𝒞𝔇+(U):={f∈C[a,b]:𝔇f(x)≥0forx∈[a,b]}.Remark 2.3.
Note that if a canonical set of bestL1-approximation exists for a functional set, then the nonlinear problem of best L1-approximation becomes a linear interpolation problem in this functional set.


### 2.3. BestL1-Approximation by Algebraic Polynomials

Denote byπn-1 the linear space of polynomials of degree ≤n-1. Taking into account that the polynomial basis  uk=xk-1,k=1,…,n  of  πn-1  is a T-system of order n on any interval [a,b], the best polynomial L1-approximant to a given function f from πn-1 can be characterized as a polynomial interpolant to f with respect to a canonical set if f belongs to an appropriately chosen convex cone of functions. In this direction we formulate a result due to S. Bernstein [2, 3].Theorem 2.4.
Letf∈Cn[-1,1] and f(n)(x)≥0 for x∈[-1,1]. Then, the polynomial pf of degree ≤n-1 is the unique polynomial of best L1-approximation to f from πn-1 on [-1,1] if and only if
(2.5)pf(xk)=f(xk),1≤k≤n,
where the interpolation nodes xk=cos[kπ/(n+1)],  k=1,…,n  are the zeros of the n-degree Chebyshev polynomial of second kind
(2.6)Un(x)=sin[(n+1)arccos(x)]1-x2.Remark 2.5.
By Theorem2.4, the interpolation points xk,k=1,…,n form a canonical set of best L1-approximation from πn-1 to the convex cone
(2.7){f∈Cn[-1,1]:f(n)(x)≥0,x∈[-1,1]}.Note that a canonical set may change with respect to the choice of the approximating space. For example, following Theorem2.4, the canonical set of best polynomial L1-approximation depends on the degree of the polynomial approximant.Corollary 2.6.
Then-degree polynomial
(2.8)Un(x)2n=sin[(n+1)arccos(x)]2n1-x2
is the unique monic (with leading coefficient 1) polynomial of degree n which has a minimal L1-norm (minimal L1-deviation) on [-1,1].We formulate a result on bestL1-approximation by algebraic polynomials according to A. Markov (see [2, 3] for details) that is based on the notion of a Haar-Chebyshev system.Theorem 2.7.
Letf∈C[-1,1] be such that the set
(2.9){1,x,x2,…,xn-1,f}
is a Haar-Chebyshev system of order n+1 on [-1,1]. Then the unique polynomial Lagrange interpolant pf to f from πn-1 with respect to the zeros xk,k=1,…,n of the n-degree Chebyshev polynomial of second kind is the unique best L1-approximant to f from πn-1.Remark 2.8.
In the particular casef∈C(n)[-1,1] and f(n)(x)>0,x∈[-1,1] of Theorem 2.4, the set of functions {1,x,2,…,xn-1,f} is a Haar-Chebyshev system of order n+1 in [-1,1]. Hence in this particular case, Theorem 2.4 is a corollary by Theorem 2.7. The notion of Haar-Chebyshev system implies that  f-p  cannot have more than n zeros on [-1,1], where p∈πn-1.


### 2.4. BestL1-Approximation by Trigonometric Polynomials

LetTn  be the linear space of trigonometric polynomials of degree ≤n. Consider the normed linear space L1[0,2π] of 2π-periodic functions whose absolute value has a finite integral on [0,2π] and equipped with the L1-norm
(2.10)∥f∥1:=∫02π|f(θ)|dθ.
The linear space Tn is a finite dimensional subspace of L1[0,2π]. For a given f∈L1[0,2π], there exists a best L1-approximant t∈Tn  (see [2] for details). Note that Tn is a Haar-Chebyshev space of order 2n+1 on [0,2π). Let C~2n+1[0,2π) denote the linear space of 2π-periodic functions having continuous derivatives of order 2n+1 on [0,2π). The next theorem [7] is a canonical set characterization of the best trigonometric L1-approximants from Tn to functions from the convex cone
(2.11)C~+2n+1:={f:f∈C~2n+1[0,2π),𝔇(2n+1)f(θ)≥0forθ∈[0,2π)},
where the differential operator 𝔇(2n+1) is defined as
(2.12)𝔇(2n+1):=D∏k=1n(D2+k2),D=ddx.Theorem 2.9.
Letf∈C~+2n+1. Then, the unique Lagrange trigonometric interpolant tf from Tn to f at the interpolation nodes kπ/(n+1),k=1,…,2n+1 is the unique best L1-approximant to f from Tn.


### 2.5. Characterization of Best One-SidedL1(μ)-Approximation

Best one-sidedL1(μ)-approximation is related to the principal representations of the measureμ, that is, to the so-called quadrature formulae of Gaussian, Lobatto, and Radau type (for details see [1, 3–6]).


#### 2.5.1. Quadrature Formulae of Gaussian, Lobatto, and Radau Type

LetU be a Haar-Chebyshev space of order n.(a)
Ifn=2m,  then there exist unique sets of points a<x1G<x2G<⋯<xmG<b  and a=x0L<x1L<⋯<xmL=b,  and unique sets of positive numbers  {ajG}j=1m and {ajL}j=0m such that the quadrature formulae
(2.13)∫abfdμ≈QG[f]:=∑j=1majGf(xjG)(Gaussiantypequadratureformula),∫abfdμ≈QL[f]:=∑j=0majLf(xjL)(Lobattotypequadratureformula)
are exact for all u∈U.(b)
Ifn=2m+1, then there exist unique sets of points  a=x0R-<x1R-<⋯<xmR-<b and  a<x0R+<x1R+<⋯<xmR+=b,  and unique sets of positive numbers{ajR-}j=0m and {ajR+}j=0m  such that the quadrature formulae
(2.14)∫abfdμ≈QR-[f]:=∑j=0majR-f(xjR-)(leftRadautypequadratureformula),∫abfdμ≈QR+[f]:=∑j=0majR+f(xjR+)(rightRadautypequadratureformula)
are exact for all u∈U.Letu1,u2,…,un be a basis of U such that  det{ui(xj)}i,j=1n>0  for  a≤x1<x2<⋯<xn≤b.  Hence, the set of functions {u1,u2,…,un} is a Haar-Chebyshev system (T-system) of order n on [a,b] or in other words U is an n-order T-space. The following theorem holds true.Theorem 2.10.
LetU⊂C[a,b] be a T-space of order n. Then, for every f∈𝒞+(U) there exist elements of best one-sided L1(μ)-approximation from below u*f∈U and from above uf*∈U. Moreover, the best one-sided approximants u*f∈U, uf*∈U to f and the function f satisfy (a)
QG[u*f]=QG[f]  and  QL[uf*]=QL[f],  if n=2m,  and(b)
QR-[u*f]=QR-[f]  and  QR+[uf*]=QR+[f],  if n=2m+1.In order to guarantee the uniqueness of the best one-sidedL1(μ)-approximant to a given function we need the following restriction on U.


#### 2.5.2. Extended Haar-Chebyshev System of Multiplicity 2

A set ofn functions {u1,u2,…,un}⊂C1[a,b] is an n-order extended Haar-Chebyshev system of multiplicity 2 (n-order ET-system of multiplicity 2) on the interval [a,b] if each nontrivial linear combination u≢0 of u1,u2,…,un has at most n-1 zeros in [a,b], provided that the common zeros of u and u′ are counted twice (counting multiplicities 2). In other words, U is an n-order extended Haar-Chebyshev space of multiplicity 2 (n-order ET-space of multiplicity 2) on the interval [a,b], if U⊂C1[a,b], dim(U)=n and each nontrivial  u∈U,u≢0 has at most n-1 zeros in [a,b], provided that the common zeros of u and u' are counted twice (counting multiplicities 2).


#### 2.5.3. Construction of Best One-SidedL1(μ)-Approximants via Interpolation on Canonical Sets

For details on the results presented here, see [1, 3, 4] and the references given there.Theorem 2.11.
LetU⊂C1[a,b] be an n-order ET-space of multiplicity 2 on the interval [a,b]. Then, for every  f∈𝒞+(U)∩C1[a,b] one has the following.(a)
Ifn=2m,  then the unique solutions  u*f and uf*  of the Lagrange-Hermite interpolation problems
(2.15)u*f(xjG)=f(xjG),u*f′(xjG)=f′(xjG),j=1,2,…,m,u*f∈U,uf*(xjL)=f(xjL),(uf*)′(xjL)=f′(xjL),j=1,2,…,m-1uf*(x0L)=f(x0L),uf*(xmL)=f(xmL),uf*∈U,
are the unique best one-sided from below, respectively, from above L1(μ)-approximants to f from U.(b)
Ifn=2m+1,  then the unique solutions  u*f and  uf* of the Lagrange-Hermite interpolation problems
(2.16)u*f(xjR-)=f(xjR-),u*f′(xjR-)=f′(xjR-),j=1,2,…,m,u*f(x0R-)=f(x0R-),u*f∈U,uf*(xjR+)=f(xjR+),(uf*)′(xjR+)=f′(xjR+),j=0,1,…,m-1,uf*(xmR+)=f(xmR+),uf*∈U,are the unique  best one-sided from below, respectively, from above  L1(μ)-approximants to f from  U.Best one-sidedL1-approximant to a given function exists and is unique under some conditions. Next theorem (for details and similar results see [3, 4, 8]) is an example in this direction.Theorem 2.12.
Letf be a differentiable, 2π-periodic function and let Tn denote the linear space of all trigonometric polynomials of degree at most n. Then, the best one-sided L1-approximant from above, respectively, from below to f from Tn exists and is unique.Remark 2.13.
The set of trigonometric polynomialst*(θ)=αcos(nθ), where -1≤α≤1  are best one-sided from below L1-approximants to the continuous 2π-periodic function f(θ)=|cos(nθ)| from Tn. Hence, continuity is not enough for uniqueness of the best one-sided L1-approximant to be claimed.The following theorem (see [7] for details) gives a characterization of the best one-sided L1-approximants via canonical sets. It can be considered as a refinement of Theorem 2.12 for functions from the convex cone  C~+2n+1.Theorem 2.14.
Letf∈C~+2n+1.  Let t*f∈Tn be the unique interpolant from Tn to f at the interpolation nodes 2kπ/(n+1),k=1,2,…,n with multiplicities 2 and t*f(0)=f(0). The interpolant t*f to f from Tn is the unique best one-sided from below L1-approximant to f from Tn.
Lettf*∈Tn be the unique interpolant from Tn to f at the interpolation nodes 2kπ/(n+1),k=1,2,…,n with multiplicities 2 and tf*(2π)=f(2π-). The interpolant tf* to f from Tn is the unique best one-sided from above L1-approximant to f from Tn.In order to proceed with the multivariate case we remark that the above-stated univariate results are constituted by the following prerequisites.(1)
The domainK where the functions are defined is an interval.(2)
The approximating spaceU in most of the cases is a kernel of a linear ordinary differential operator 𝔇, that is,  U=Ker𝔇.(3)
The set on which certain interpolation problem is unisolvent forU consists of a finite number of points.(4)
The convex cone𝒞𝔇+(U) of functions to be approximated is defined by
(2.17)𝒞𝔇+(U):={f:𝔇fexistsand𝔇f≥0on[a,b]}.(5)
The best (one-sided)L1-approximants are characterized as Lagrange-Hermite interpolants on canonical sets.Conditions(3)–(5) are due to the fact that U is a T-space or ET-space of multiplicity 2. It is known that there are no point-wise Haar-Chebyshev systems in the multivariate case [9, 10] provided that the interpolation set is of finite number of points and the interpolating space U is finite-dimensional. This fact is not surprising taking into account that the kernels of linear partial differential operators are infinite-dimensional linear spaces. Therefore, the natural interpolation sets in the multivariate case K⊂Rd should be nondenumerable point sets, and more precisely, (d-1)-dimensional manifolds. Hence, an appropriate transfinite interpolation on lower dimensional manifolds can be a basis for a canonical set characterization of the best L1-approximants in the multivariate case. However, unlike the finite number of points in an interval [a,b] which are topologically equivalent, there is a countless variety of possible interpolation sets (interpolation grids) in the multivariate case and one cannot expect that there would be a simple characterization corresponding to that in the univariate case. Instead, one should consider pairs(U,X) consisting of an approximation space U and a set X⊂K such that X is unisolvent for U. Such sets X are candidates for canonical sets in the multivariate L1-approximation. There is no consistent general treatment of the canonical sets of best approximation in Rd and each known result is a solution of a particular problem. In the next sections we will discuss some of these results and we will focus on multivariate interpolation and best L1-approximation by blending functions.


## 2.1. Haar-Chebyshev System of Ordern

A set ofn functions {u1,u2,…,un}⊂C[a,b] is a Haar-Chebyshev system (T-system) of order n on the interval [a,b] if each nontrivial linear combination of u1,u2,…,un has at most n-1 zeros in  [a,b].  In other words, a set of n functions {u1,u2,…,un} is a Haar-Chebyshev system of order n on the interval [a,b] if it is linearly independent with respect to an arbitrary chosen set of n distinct points in [a,b]. We say that an n-dimensional subspace U of  C[a,b] is a Haar-Chebyshev space(T-space)of ordern on [a,b] if every  u∈U, u≢0, has no more than n-1 distinct zeros in [a,b]. Equivalently, U is a T-space of order n on [a,b], if for every basis u1,u2,…,un of U there exists an ε∈{-1,1} such that εdet{ui(xj)}i,j=1n>0  for all  a≤x1<x2<⋯<xn≤b. It is also often said that the basis functions u1,u2,…,un constitute a T-space of order n on [a,b].


## 2.2. Characterization of BestL1(μ)-Approximation by Canonical Interpolation Sets

Following [1], throughout this section we suppose that μ is a finite positive nonatomic measure. The problem of best L1(μ)-approximation by a Haar-Chebyshev space has an elegant solution via interpolation on a canonical set. Let U⊂C[a,b] be a Haar-Chebyshev space of order n and let u1,u2,…,un be a basis of U. Consider the convex cone 𝒞+(U) defined as
(2.1)𝒞+(U):={un+1:det{ui(xj)}i,j=1n+1≥0}
for all a≤x1<x2<⋯<xn<xn+1≤b.Theorem 2.1.
LetU⊂C[a,b] be a Haar-Chebyshev space of order n on [a,b]. Then the following holds true.(a)
Uniqueness of the Canonical Set. There is a unique set of pointsa=x0<x1<⋯<xn+1=b such that
(2.2)∑j=1n+1(-1)j∫xj-1xjudμ=0,∀u∈U.(b)
BestL1(μ)-Approximation via Interpolation. Let f∈𝒞+(U). Then the unique solution uf∈U of the interpolation problem
(2.3)uf(xj)=f(xj),j=1,2,…,n
is the unique best L1(μ)-approximant to f from U.Remark 2.2.
Following Theorem2.1, the set {xk}k=1n is a canonical set of best L1(μ)-approximation to the convex cone 𝒞+(U) of functions from the T-space U. In many cases the approximating subspace U can be a kernel of a certain differential operator 𝔇, that is, U=Ker𝔇 and the convex cone 𝒞𝔇+(U) of functions to be approximated can be defined by
(2.4)𝒞𝔇+(U):={f∈C[a,b]:𝔇f(x)≥0forx∈[a,b]}.Remark 2.3.
Note that if a canonical set of bestL1-approximation exists for a functional set, then the nonlinear problem of best L1-approximation becomes a linear interpolation problem in this functional set.


## 2.3. BestL1-Approximation by Algebraic Polynomials

Denote byπn-1 the linear space of polynomials of degree ≤n-1. Taking into account that the polynomial basis  uk=xk-1,k=1,…,n  of  πn-1  is a T-system of order n on any interval [a,b], the best polynomial L1-approximant to a given function f from πn-1 can be characterized as a polynomial interpolant to f with respect to a canonical set if f belongs to an appropriately chosen convex cone of functions. In this direction we formulate a result due to S. Bernstein [2, 3].Theorem 2.4.
Letf∈Cn[-1,1] and f(n)(x)≥0 for x∈[-1,1]. Then, the polynomial pf of degree ≤n-1 is the unique polynomial of best L1-approximation to f from πn-1 on [-1,1] if and only if
(2.5)pf(xk)=f(xk),1≤k≤n,
where the interpolation nodes xk=cos[kπ/(n+1)],  k=1,…,n  are the zeros of the n-degree Chebyshev polynomial of second kind
(2.6)Un(x)=sin[(n+1)arccos(x)]1-x2.Remark 2.5.
By Theorem2.4, the interpolation points xk,k=1,…,n form a canonical set of best L1-approximation from πn-1 to the convex cone
(2.7){f∈Cn[-1,1]:f(n)(x)≥0,x∈[-1,1]}.Note that a canonical set may change with respect to the choice of the approximating space. For example, following Theorem2.4, the canonical set of best polynomial L1-approximation depends on the degree of the polynomial approximant.Corollary 2.6.
Then-degree polynomial
(2.8)Un(x)2n=sin[(n+1)arccos(x)]2n1-x2
is the unique monic (with leading coefficient 1) polynomial of degree n which has a minimal L1-norm (minimal L1-deviation) on [-1,1].We formulate a result on bestL1-approximation by algebraic polynomials according to A. Markov (see [2, 3] for details) that is based on the notion of a Haar-Chebyshev system.Theorem 2.7.
Letf∈C[-1,1] be such that the set
(2.9){1,x,x2,…,xn-1,f}
is a Haar-Chebyshev system of order n+1 on [-1,1]. Then the unique polynomial Lagrange interpolant pf to f from πn-1 with respect to the zeros xk,k=1,…,n of the n-degree Chebyshev polynomial of second kind is the unique best L1-approximant to f from πn-1.Remark 2.8.
In the particular casef∈C(n)[-1,1] and f(n)(x)>0,x∈[-1,1] of Theorem 2.4, the set of functions {1,x,2,…,xn-1,f} is a Haar-Chebyshev system of order n+1 in [-1,1]. Hence in this particular case, Theorem 2.4 is a corollary by Theorem 2.7. The notion of Haar-Chebyshev system implies that  f-p  cannot have more than n zeros on [-1,1], where p∈πn-1.


## 2.4. BestL1-Approximation by Trigonometric Polynomials

LetTn  be the linear space of trigonometric polynomials of degree ≤n. Consider the normed linear space L1[0,2π] of 2π-periodic functions whose absolute value has a finite integral on [0,2π] and equipped with the L1-norm
(2.10)∥f∥1:=∫02π|f(θ)|dθ.
The linear space Tn is a finite dimensional subspace of L1[0,2π]. For a given f∈L1[0,2π], there exists a best L1-approximant t∈Tn  (see [2] for details). Note that Tn is a Haar-Chebyshev space of order 2n+1 on [0,2π). Let C~2n+1[0,2π) denote the linear space of 2π-periodic functions having continuous derivatives of order 2n+1 on [0,2π). The next theorem [7] is a canonical set characterization of the best trigonometric L1-approximants from Tn to functions from the convex cone
(2.11)C~+2n+1:={f:f∈C~2n+1[0,2π),𝔇(2n+1)f(θ)≥0forθ∈[0,2π)},
where the differential operator 𝔇(2n+1) is defined as
(2.12)𝔇(2n+1):=D∏k=1n(D2+k2),D=ddx.Theorem 2.9.
Letf∈C~+2n+1. Then, the unique Lagrange trigonometric interpolant tf from Tn to f at the interpolation nodes kπ/(n+1),k=1,…,2n+1 is the unique best L1-approximant to f from Tn.


## 2.5. Characterization of Best One-SidedL1(μ)-Approximation

Best one-sidedL1(μ)-approximation is related to the principal representations of the measureμ, that is, to the so-called quadrature formulae of Gaussian, Lobatto, and Radau type (for details see [1, 3–6]).


### 2.5.1. Quadrature Formulae of Gaussian, Lobatto, and Radau Type

LetU be a Haar-Chebyshev space of order n.(a)
Ifn=2m,  then there exist unique sets of points a<x1G<x2G<⋯<xmG<b  and a=x0L<x1L<⋯<xmL=b,  and unique sets of positive numbers  {ajG}j=1m and {ajL}j=0m such that the quadrature formulae
(2.13)∫abfdμ≈QG[f]:=∑j=1majGf(xjG)(Gaussiantypequadratureformula),∫abfdμ≈QL[f]:=∑j=0majLf(xjL)(Lobattotypequadratureformula)
are exact for all u∈U.(b)
Ifn=2m+1, then there exist unique sets of points  a=x0R-<x1R-<⋯<xmR-<b and  a<x0R+<x1R+<⋯<xmR+=b,  and unique sets of positive numbers{ajR-}j=0m and {ajR+}j=0m  such that the quadrature formulae
(2.14)∫abfdμ≈QR-[f]:=∑j=0majR-f(xjR-)(leftRadautypequadratureformula),∫abfdμ≈QR+[f]:=∑j=0majR+f(xjR+)(rightRadautypequadratureformula)
are exact for all u∈U.Letu1,u2,…,un be a basis of U such that  det{ui(xj)}i,j=1n>0  for  a≤x1<x2<⋯<xn≤b.  Hence, the set of functions {u1,u2,…,un} is a Haar-Chebyshev system (T-system) of order n on [a,b] or in other words U is an n-order T-space. The following theorem holds true.Theorem 2.10.
LetU⊂C[a,b] be a T-space of order n. Then, for every f∈𝒞+(U) there exist elements of best one-sided L1(μ)-approximation from below u*f∈U and from above uf*∈U. Moreover, the best one-sided approximants u*f∈U, uf*∈U to f and the function f satisfy (a)
QG[u*f]=QG[f]  and  QL[uf*]=QL[f],  if n=2m,  and(b)
QR-[u*f]=QR-[f]  and  QR+[uf*]=QR+[f],  if n=2m+1.In order to guarantee the uniqueness of the best one-sidedL1(μ)-approximant to a given function we need the following restriction on U.


### 2.5.2. Extended Haar-Chebyshev System of Multiplicity 2

A set ofn functions {u1,u2,…,un}⊂C1[a,b] is an n-order extended Haar-Chebyshev system of multiplicity 2 (n-order ET-system of multiplicity 2) on the interval [a,b] if each nontrivial linear combination u≢0 of u1,u2,…,un has at most n-1 zeros in [a,b], provided that the common zeros of u and u′ are counted twice (counting multiplicities 2). In other words, U is an n-order extended Haar-Chebyshev space of multiplicity 2 (n-order ET-space of multiplicity 2) on the interval [a,b], if U⊂C1[a,b], dim(U)=n and each nontrivial  u∈U,u≢0 has at most n-1 zeros in [a,b], provided that the common zeros of u and u' are counted twice (counting multiplicities 2).


### 2.5.3. Construction of Best One-SidedL1(μ)-Approximants via Interpolation on Canonical Sets

For details on the results presented here, see [1, 3, 4] and the references given there.Theorem 2.11.
LetU⊂C1[a,b] be an n-order ET-space of multiplicity 2 on the interval [a,b]. Then, for every  f∈𝒞+(U)∩C1[a,b] one has the following.(a)
Ifn=2m,  then the unique solutions  u*f and uf*  of the Lagrange-Hermite interpolation problems
(2.15)u*f(xjG)=f(xjG),u*f′(xjG)=f′(xjG),j=1,2,…,m,u*f∈U,uf*(xjL)=f(xjL),(uf*)′(xjL)=f′(xjL),j=1,2,…,m-1uf*(x0L)=f(x0L),uf*(xmL)=f(xmL),uf*∈U,
are the unique best one-sided from below, respectively, from above L1(μ)-approximants to f from U.(b)
Ifn=2m+1,  then the unique solutions  u*f and  uf* of the Lagrange-Hermite interpolation problems
(2.16)u*f(xjR-)=f(xjR-),u*f′(xjR-)=f′(xjR-),j=1,2,…,m,u*f(x0R-)=f(x0R-),u*f∈U,uf*(xjR+)=f(xjR+),(uf*)′(xjR+)=f′(xjR+),j=0,1,…,m-1,uf*(xmR+)=f(xmR+),uf*∈U,are the unique  best one-sided from below, respectively, from above  L1(μ)-approximants to f from  U.Best one-sidedL1-approximant to a given function exists and is unique under some conditions. Next theorem (for details and similar results see [3, 4, 8]) is an example in this direction.Theorem 2.12.
Letf be a differentiable, 2π-periodic function and let Tn denote the linear space of all trigonometric polynomials of degree at most n. Then, the best one-sided L1-approximant from above, respectively, from below to f from Tn exists and is unique.Remark 2.13.
The set of trigonometric polynomialst*(θ)=αcos(nθ), where -1≤α≤1  are best one-sided from below L1-approximants to the continuous 2π-periodic function f(θ)=|cos(nθ)| from Tn. Hence, continuity is not enough for uniqueness of the best one-sided L1-approximant to be claimed.The following theorem (see [7] for details) gives a characterization of the best one-sided L1-approximants via canonical sets. It can be considered as a refinement of Theorem 2.12 for functions from the convex cone  C~+2n+1.Theorem 2.14.
Letf∈C~+2n+1.  Let t*f∈Tn be the unique interpolant from Tn to f at the interpolation nodes 2kπ/(n+1),k=1,2,…,n with multiplicities 2 and t*f(0)=f(0). The interpolant t*f to f from Tn is the unique best one-sided from below L1-approximant to f from Tn.
Lettf*∈Tn be the unique interpolant from Tn to f at the interpolation nodes 2kπ/(n+1),k=1,2,…,n with multiplicities 2 and tf*(2π)=f(2π-). The interpolant tf* to f from Tn is the unique best one-sided from above L1-approximant to f from Tn.In order to proceed with the multivariate case we remark that the above-stated univariate results are constituted by the following prerequisites.(1)
The domainK where the functions are defined is an interval.(2)
The approximating spaceU in most of the cases is a kernel of a linear ordinary differential operator 𝔇, that is,  U=Ker𝔇.(3)
The set on which certain interpolation problem is unisolvent forU consists of a finite number of points.(4)
The convex cone𝒞𝔇+(U) of functions to be approximated is defined by
(2.17)𝒞𝔇+(U):={f:𝔇fexistsand𝔇f≥0on[a,b]}.(5)
The best (one-sided)L1-approximants are characterized as Lagrange-Hermite interpolants on canonical sets.Conditions(3)–(5) are due to the fact that U is a T-space or ET-space of multiplicity 2. It is known that there are no point-wise Haar-Chebyshev systems in the multivariate case [9, 10] provided that the interpolation set is of finite number of points and the interpolating space U is finite-dimensional. This fact is not surprising taking into account that the kernels of linear partial differential operators are infinite-dimensional linear spaces. Therefore, the natural interpolation sets in the multivariate case K⊂Rd should be nondenumerable point sets, and more precisely, (d-1)-dimensional manifolds. Hence, an appropriate transfinite interpolation on lower dimensional manifolds can be a basis for a canonical set characterization of the best L1-approximants in the multivariate case. However, unlike the finite number of points in an interval [a,b] which are topologically equivalent, there is a countless variety of possible interpolation sets (interpolation grids) in the multivariate case and one cannot expect that there would be a simple characterization corresponding to that in the univariate case. Instead, one should consider pairs(U,X) consisting of an approximation space U and a set X⊂K such that X is unisolvent for U. Such sets X are candidates for canonical sets in the multivariate L1-approximation. There is no consistent general treatment of the canonical sets of best approximation in Rd and each known result is a solution of a particular problem. In the next sections we will discuss some of these results and we will focus on multivariate interpolation and best L1-approximation by blending functions.


## 2.5.1. Quadrature Formulae of Gaussian, Lobatto, and Radau Type

LetU be a Haar-Chebyshev space of order n.(a)
Ifn=2m,  then there exist unique sets of points a<x1G<x2G<⋯<xmG<b  and a=x0L<x1L<⋯<xmL=b,  and unique sets of positive numbers  {ajG}j=1m and {ajL}j=0m such that the quadrature formulae
(2.13)∫abfdμ≈QG[f]:=∑j=1majGf(xjG)(Gaussiantypequadratureformula),∫abfdμ≈QL[f]:=∑j=0majLf(xjL)(Lobattotypequadratureformula)
are exact for all u∈U.(b)
Ifn=2m+1, then there exist unique sets of points  a=x0R-<x1R-<⋯<xmR-<b and  a<x0R+<x1R+<⋯<xmR+=b,  and unique sets of positive numbers{ajR-}j=0m and {ajR+}j=0m  such that the quadrature formulae
(2.14)∫abfdμ≈QR-[f]:=∑j=0majR-f(xjR-)(leftRadautypequadratureformula),∫abfdμ≈QR+[f]:=∑j=0majR+f(xjR+)(rightRadautypequadratureformula)
are exact for all u∈U.Letu1,u2,…,un be a basis of U such that  det{ui(xj)}i,j=1n>0  for  a≤x1<x2<⋯<xn≤b.  Hence, the set of functions {u1,u2,…,un} is a Haar-Chebyshev system (T-system) of order n on [a,b] or in other words U is an n-order T-space. The following theorem holds true.Theorem 2.10.
LetU⊂C[a,b] be a T-space of order n. Then, for every f∈𝒞+(U) there exist elements of best one-sided L1(μ)-approximation from below u*f∈U and from above uf*∈U. Moreover, the best one-sided approximants u*f∈U, uf*∈U to f and the function f satisfy (a)
QG[u*f]=QG[f]  and  QL[uf*]=QL[f],  if n=2m,  and(b)
QR-[u*f]=QR-[f]  and  QR+[uf*]=QR+[f],  if n=2m+1.In order to guarantee the uniqueness of the best one-sidedL1(μ)-approximant to a given function we need the following restriction on U.


## 2.5.2. Extended Haar-Chebyshev System of Multiplicity 2

A set ofn functions {u1,u2,…,un}⊂C1[a,b] is an n-order extended Haar-Chebyshev system of multiplicity 2 (n-order ET-system of multiplicity 2) on the interval [a,b] if each nontrivial linear combination u≢0 of u1,u2,…,un has at most n-1 zeros in [a,b], provided that the common zeros of u and u′ are counted twice (counting multiplicities 2). In other words, U is an n-order extended Haar-Chebyshev space of multiplicity 2 (n-order ET-space of multiplicity 2) on the interval [a,b], if U⊂C1[a,b], dim(U)=n and each nontrivial  u∈U,u≢0 has at most n-1 zeros in [a,b], provided that the common zeros of u and u' are counted twice (counting multiplicities 2).


## 2.5.3. Construction of Best One-SidedL1(μ)-Approximants via Interpolation on Canonical Sets

For details on the results presented here, see [1, 3, 4] and the references given there.Theorem 2.11.
LetU⊂C1[a,b] be an n-order ET-space of multiplicity 2 on the interval [a,b]. Then, for every  f∈𝒞+(U)∩C1[a,b] one has the following.(a)
Ifn=2m,  then the unique solutions  u*f and uf*  of the Lagrange-Hermite interpolation problems
(2.15)u*f(xjG)=f(xjG),u*f′(xjG)=f′(xjG),j=1,2,…,m,u*f∈U,uf*(xjL)=f(xjL),(uf*)′(xjL)=f′(xjL),j=1,2,…,m-1uf*(x0L)=f(x0L),uf*(xmL)=f(xmL),uf*∈U,
are the unique best one-sided from below, respectively, from above L1(μ)-approximants to f from U.(b)
Ifn=2m+1,  then the unique solutions  u*f and  uf* of the Lagrange-Hermite interpolation problems
(2.16)u*f(xjR-)=f(xjR-),u*f′(xjR-)=f′(xjR-),j=1,2,…,m,u*f(x0R-)=f(x0R-),u*f∈U,uf*(xjR+)=f(xjR+),(uf*)′(xjR+)=f′(xjR+),j=0,1,…,m-1,uf*(xmR+)=f(xmR+),uf*∈U,are the unique  best one-sided from below, respectively, from above  L1(μ)-approximants to f from  U.Best one-sidedL1-approximant to a given function exists and is unique under some conditions. Next theorem (for details and similar results see [3, 4, 8]) is an example in this direction.Theorem 2.12.
Letf be a differentiable, 2π-periodic function and let Tn denote the linear space of all trigonometric polynomials of degree at most n. Then, the best one-sided L1-approximant from above, respectively, from below to f from Tn exists and is unique.Remark 2.13.
The set of trigonometric polynomialst*(θ)=αcos(nθ), where -1≤α≤1  are best one-sided from below L1-approximants to the continuous 2π-periodic function f(θ)=|cos(nθ)| from Tn. Hence, continuity is not enough for uniqueness of the best one-sided L1-approximant to be claimed.The following theorem (see [7] for details) gives a characterization of the best one-sided L1-approximants via canonical sets. It can be considered as a refinement of Theorem 2.12 for functions from the convex cone  C~+2n+1.Theorem 2.14.
Letf∈C~+2n+1.  Let t*f∈Tn be the unique interpolant from Tn to f at the interpolation nodes 2kπ/(n+1),k=1,2,…,n with multiplicities 2 and t*f(0)=f(0). The interpolant t*f to f from Tn is the unique best one-sided from below L1-approximant to f from Tn.
Lettf*∈Tn be the unique interpolant from Tn to f at the interpolation nodes 2kπ/(n+1),k=1,2,…,n with multiplicities 2 and tf*(2π)=f(2π-). The interpolant tf* to f from Tn is the unique best one-sided from above L1-approximant to f from Tn.In order to proceed with the multivariate case we remark that the above-stated univariate results are constituted by the following prerequisites.(1)
The domainK where the functions are defined is an interval.(2)
The approximating spaceU in most of the cases is a kernel of a linear ordinary differential operator 𝔇, that is,  U=Ker𝔇.(3)
The set on which certain interpolation problem is unisolvent forU consists of a finite number of points.(4)
The convex cone𝒞𝔇+(U) of functions to be approximated is defined by
(2.17)𝒞𝔇+(U):={f:𝔇fexistsand𝔇f≥0on[a,b]}.(5)
The best (one-sided)L1-approximants are characterized as Lagrange-Hermite interpolants on canonical sets.Conditions(3)–(5) are due to the fact that U is a T-space or ET-space of multiplicity 2. It is known that there are no point-wise Haar-Chebyshev systems in the multivariate case [9, 10] provided that the interpolation set is of finite number of points and the interpolating space U is finite-dimensional. This fact is not surprising taking into account that the kernels of linear partial differential operators are infinite-dimensional linear spaces. Therefore, the natural interpolation sets in the multivariate case K⊂Rd should be nondenumerable point sets, and more precisely, (d-1)-dimensional manifolds. Hence, an appropriate transfinite interpolation on lower dimensional manifolds can be a basis for a canonical set characterization of the best L1-approximants in the multivariate case. However, unlike the finite number of points in an interval [a,b] which are topologically equivalent, there is a countless variety of possible interpolation sets (interpolation grids) in the multivariate case and one cannot expect that there would be a simple characterization corresponding to that in the univariate case. Instead, one should consider pairs(U,X) consisting of an approximation space U and a set X⊂K such that X is unisolvent for U. Such sets X are candidates for canonical sets in the multivariate L1-approximation. There is no consistent general treatment of the canonical sets of best approximation in Rd and each known result is a solution of a particular problem. In the next sections we will discuss some of these results and we will focus on multivariate interpolation and best L1-approximation by blending functions.


## 3. Algebraic Blending Functions

The linear spaceπm-1 of univariate polynomials of degree ≤m-1 can be defined as the general solution of the homogeneous, m-order, linear differential equation p(m)(x)=0,x∈[-1,1]. Hence,
(3.1)πm-1={p∈Cm(I):dmpdxm=0,I:=[-1,1]},
where by Cm(I) we denote the linear space of functions having continuous m-order derivative on [-1,1]. The linear space πm-1 is of finite dimension m.LetI2=[-1,1]2 and
(3.2)Cm,n:={f(x,y):Di,jf(x,y)∈C(I2),0≤i≤m,0≤j≤n},
where  Di,jf(x,y):=(∂i+j/∂xi∂yj)f(x,y) is the partial derivative of f of order (i,j). Cm,n denotes the linear space of all real valued functions having continuous partial derivatives up to order  (m,n) on  I2.The classical univariate polynomials have a natural multivariate extension by the so-calledalgebraic blending functions. Let m,n be nonnegative integer such that m+n≥1.  The space Bm,n of algebraic blending functions of order (m,n) on the unit square I2 is defined as
(3.3)Bm,n:={h∈Cm,n:Dm,nh(x,y)=0,(x,y)∈I2}.
In other words, the linear space Bm,n is the general solution of the homogeneous, (m,n)-order, linear partial differential equation Dm,nh(x,y)=0 on I2 or, saying it differently, the linear space Bm,n is the kernel of the linear partial differential operator Dm,n.Each blending functionh∈Bm,n of order (m,n) can be represented in the form
(3.4)h(x,y)=∑k=0m-1ak(y)xk+∑l=0n-1bl(x)yl,ak(y)∈Cn(I),bl(x)∈Cm(I)
and obviously, each such function belongs to Bm,n. Hence,
(3.5)Bm,n={h:h(x,y)=∑k=0m-1ak(y)xk+∑l=0n-1bl(x)yl,ak∈Cn(I),bl∈Cm(I)}.
Note that the above representation of a function from Bm,n is not unique and the linear space Bm,n is of infinite dimension.


## 4. Transfinite Interpolation by Algebraic Blending Functions

In two-dimensional case, transfinite interpolation (beyond the finite interpolation) is to construct a simple interpolation function (blending interpolation function for example) over a planar domain in such a way that it matches (interpolates) a given function and its partial derivatives on curves. Transfinite interpolation is an approximate recovery of functions via interpolation with variety of applications, for example, in geometric modeling and finite element methods. The notion can be extended in a natural way for higher dimensions. The Dirichlet problem is an example of a transfinite interpolation scheme in the linear space of harmonic functions.In contrast to the transfinite interpolation schemes, the classical interpolation schemes are restricted to a finite or denumerable number of interpolation points. For example, approximate recovery of a univariate function can be obtained by a univariate Lagrange interpolation polynomial: letf∈C[-1,1] and x1,x2,…,xm be m distinct points in [-1,1]. Then, there exists a unique polynomial interpolant pf∈πm-1 to f of degree ≤m-1 such that
(4.1)pf(xk)=f(xk),k=1,2,…,m
with an error representation formula
(4.2)f(x)-pf(x)=f(m)(ξ)m!(x-x1)⋯(x-xm),x∈[-1,1],ξ∈(-1,1).
Existence and uniqueness of univariate Lagrange interpolants are closely connected with the notion of Haar-Chebyshev system in the univariate case.


### 4.1. Haar-Chebyshev Systems in the Multivariate Case

Let us remind that a set ofm univariate functions {u1,u2,…,um}⊂C[-1,1] is a Haar-Chebyshev system (T-system) of order m in [-1,1] if each nontrivial linear combination of u1,u2,…,um has at most m-1 zeros in [-1,1]. In other words, a set of m functions is a Haar-Chebyshev system of order m in [-1,1] if it is linearly independent with respect to an arbitrary chosen set of m distinct points in [-1,1]. The concept of a Haar-Chebyshev system (T-system) is based on counting the zeros and it is essentially restricted to the univariate case. It is known [9, 10] that there is no universal Haar-Chebyshev system on any set in Rd that contains an interior point, in particular, on I2. Hence, the point-wise Lagrange univariate interpolation scheme cannot be extended to the point-wise interpolation scheme in the multivariate case.However, we can extend the notion of a Haar-Chebyshev system in the multivariate case with respect to transfinite interpolation by blending functions on grids (curves) with a prescribed geometry. Being defined as the kernel of the differential operatorDm,n, the linear space Bm,n of algebraic blending functions of order (m,n) shares interpolation properties of the algebraic polynomials. In particular, Lagrange interpolation by algebraic polynomials has a multivariate extension to transfinite Lagrange interpolation by blending functions.


### 4.2. Transfinite Interpolation Grid

Let(4.3)ωxm,yn(x,y):=∏s=1m(x-xs)∏k=1n(y-yk).
We define an (m,n) interpolation grid  Gm,n  associated with the 2 point sets of distinct in each set points  xm:={xs,1≤s≤m}  and   yn:={yk,1≤k≤n}:
(4.4)Gm,n:={(x,y)∈I2:ωxm,yn(x,y)=0}.
The interpolation grid  Gm,n  is a set of m vertical and n  horizontal line-segments in  I2.


### 4.3. Transfinite Lagrange Interpolation by Algebraic Blending Functions

Letf∈Cm,n. Let  xm:={xs,1≤s≤m} and yn:={yk,1≤k≤n}  be sets of distinct points (in each case) in  I. Then there exists a unique blending interpolant  hf∈Bm,n to f,  satisfying the following transfinite interpolation conditions:
(4.5)(hf)∣Gm,n=f∣Gm,n.
By using the  univariate fundamental Lagrange interpolating polynomials ls,m(x),s=1,…,m and lk,n(y),k=1,…,n we give the following explicit construction of the unique transfinite Lagrange interpolant to  f  on the grid Gm,n:
(4.6)hf(x,y)=∑s=1mf(xs,y)ls,m(x)+∑k=1nf(x,yk)lk,n(y)-∑s=1m∑k=1nf(xs,yk)ls,m(x)lk,n(y)
with the error representation formula
(4.7)f(x,y)-hf(x,y)=Dm,nf(ξ,η)m!n!ωxm,yn(x,y),(x,y)∈I2,(ξ,η)∈I2.
From the above error representation formula, for each g∈Bm,n we have hg=g to conclude that for a fixed (m,n) interpolation grid Gm,n there exists a unique transfinite interpolant hf from Bm,n to a given function f∈Cm,n.In addition, the univariate concept of a Haar-Chebyshev system can be extended in the infinite dimensional linear space of two-variable blending functions and interpolation grids consisting of vertical and horizontal line-segments as follows:ifh∈Bm,nsatisfiesh∣Gm,n=0,  thenh=0onI2. In other words, an(m,n)blending grid cannot be a zero set ofh∈Bm,n,hbeing nonidentically zero onI2.


### 4.4. Haar-Chebyshev Pair of a Linear Space and a Set of Grids in the Multivariate Case

In the one-dimensional Lagrange interpolation problem we have a finite set of points which are similar in the sense they obey the natural ordering on the real line. In the bivariate case we have set of grid-lines which are also similar to each other in the sense thatm of them are parallel to the one axis, and n of them, to the other. The points where the lines intersect the coordinate axes are also naturally ordered. Each blending grid Gm,n is unisolvent in transfinite Lagrange interpolation sense for Bm,n. On the other hand, the interpolation sets in the bivariate (multivariate) case may have diverse geometry and we can not talk about Haar-Chebyshev systems in general. However, we can argue that the pair of the linear space Bm,n and the set of blending grids 𝒢m,n(4.8){Bm,n,𝒢m,n}isaHaar-ChebyshevpaironI2,
where  𝒢m,n is the set of all blending grids of order (m,n) in I2, consisting of m vertical and n horizontal line-segments.


### 4.5. Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions

Let(4.9)Gm,n:={(x,y)∈I2:ωxm,yn(x,y)=0}
be (m,n) interpolation grid associated with the point sets xm:={xμ,1≤μ≤m} and yn:={yν,1≤ν≤n}. The interpolation grid Gm,n consists of m vertical and n horizontal line segments on I2. We associate with each point xμ a multiplicity sμ and with each point yν a multiplicity jν. Let ∑μ=1msμ=M and  ∑ν=1njν=N. Then for a given function f∈CM,N, there exists a unique transfinite Lagrange-Hermite blending interpolant hf from BM,N to f, satisfying the transfinite interpolation conditions
(4.10)Ds,0hf(xμ,y)=Ds,0f(xμ,y),s=0,…,sμ-1,μ=1,…,m,y∈[-1,1],D0,jhf(x,yν)=D0,jf(x,yν),j=0,…,jν-1,ν=1,…,n,x∈[-1,1]
with respect to the given (m,n) blending grid Gm,n, where as usual D0,0f=f.Explicit construction of the transfinite Lagrange-Hermite interpolanthf to f from BM,N is
(4.11)hf(x,y)=∑μ=1m∑s=0sμ-1Ds,0f(xμ,y)lμ,Ms(x)+∑ν=1n∑j=0jν-1D0,jf(x,yν)lν,Nj(y)-∑μ=1m∑s=0sμ-1∑ν=1n∑j=0jν-1Ds,jf(xμ,yν)lμ,Ms(x)lν,Nj(y),
where the fundamental Lagrange-Hermite interpolation polynomials lμ,Ms(x),μ=1,…,m and  lν,Nj(y),ν=1,…,n satisfy the interpolation conditions
(4.12)Dr,0lμ,Ms(xκ)=δμ,κδs,r,κ=1,…,m,r=0,…,sμ-1,D0,rlν,Nj(yκ)=δν,κδj,r,κ=1,…,n,r=0,…,jν-1.


### 4.6. Error Representation of Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions in Terms of B-Splines

Let(4.13)Ω(xm,sm);(yn,jn)(x,y):=∏μ=1m(x-xμ)sμ∏ν=1n(y-yν)jν,
where sm:=(s1,…,sm) and jn:=(j1,…,jn). Then the following error representation formula holds
(4.14)f(x,y)-hf(x,y)=Ω(xm,sm);(yn,jn)(x,y)×∬-11BM-1;[(xm,sm),x](v)BN-1;[(yn,jn),x](w)DM,Nf(v,w)dvdw,
where BM-1;[(xm,sm),x] is the normalized B-spline of degree M-1 with knots (xm,x) of multiplicities (sm,1) and BN-1;[(yn,jn),y] is the normalized B-spline of degree N-1 with knots (yn,y) of multiplicities (jn,1). For details on B-splines see [11–13]. As a corollary we obtain Cauchy error representation for transfinite Lagrange-Hermite interpolation by blending functions
(4.15)f(x,y)-hf(x,y)=DM,Nf(ξ,η)M!N!Ω(xm,sm);(yn,jn)(x,y),(x,y)∈I2,(ξ,η)∈I2.
Note that hg=g for a blending function g∈BM,N. Therefore, each blending function from BM,N can be represented as transfinite Lagrange-Hermite interpolant on a grid consisting of horizontal and vertical line-segments.Remark 4.1.
The multivariate results which resemble the broadest extent the univariate theory concern transfinite interpolation and bestL1-approximation by algebraic blending functions. For simplicity of the notations the results on transfinite interpolation and best L1-approximation by algebraic blending functions are stated in the bivariate case d=2 although they are entirely valid in higher dimensions.


## 4.1. Haar-Chebyshev Systems in the Multivariate Case

Let us remind that a set ofm univariate functions {u1,u2,…,um}⊂C[-1,1] is a Haar-Chebyshev system (T-system) of order m in [-1,1] if each nontrivial linear combination of u1,u2,…,um has at most m-1 zeros in [-1,1]. In other words, a set of m functions is a Haar-Chebyshev system of order m in [-1,1] if it is linearly independent with respect to an arbitrary chosen set of m distinct points in [-1,1]. The concept of a Haar-Chebyshev system (T-system) is based on counting the zeros and it is essentially restricted to the univariate case. It is known [9, 10] that there is no universal Haar-Chebyshev system on any set in Rd that contains an interior point, in particular, on I2. Hence, the point-wise Lagrange univariate interpolation scheme cannot be extended to the point-wise interpolation scheme in the multivariate case.However, we can extend the notion of a Haar-Chebyshev system in the multivariate case with respect to transfinite interpolation by blending functions on grids (curves) with a prescribed geometry. Being defined as the kernel of the differential operatorDm,n, the linear space Bm,n of algebraic blending functions of order (m,n) shares interpolation properties of the algebraic polynomials. In particular, Lagrange interpolation by algebraic polynomials has a multivariate extension to transfinite Lagrange interpolation by blending functions.


## 4.2. Transfinite Interpolation Grid

Let(4.3)ωxm,yn(x,y):=∏s=1m(x-xs)∏k=1n(y-yk).
We define an (m,n) interpolation grid  Gm,n  associated with the 2 point sets of distinct in each set points  xm:={xs,1≤s≤m}  and   yn:={yk,1≤k≤n}:
(4.4)Gm,n:={(x,y)∈I2:ωxm,yn(x,y)=0}.
The interpolation grid  Gm,n  is a set of m vertical and n  horizontal line-segments in  I2.


## 4.3. Transfinite Lagrange Interpolation by Algebraic Blending Functions

Letf∈Cm,n. Let  xm:={xs,1≤s≤m} and yn:={yk,1≤k≤n}  be sets of distinct points (in each case) in  I. Then there exists a unique blending interpolant  hf∈Bm,n to f,  satisfying the following transfinite interpolation conditions:
(4.5)(hf)∣Gm,n=f∣Gm,n.
By using the  univariate fundamental Lagrange interpolating polynomials ls,m(x),s=1,…,m and lk,n(y),k=1,…,n we give the following explicit construction of the unique transfinite Lagrange interpolant to  f  on the grid Gm,n:
(4.6)hf(x,y)=∑s=1mf(xs,y)ls,m(x)+∑k=1nf(x,yk)lk,n(y)-∑s=1m∑k=1nf(xs,yk)ls,m(x)lk,n(y)
with the error representation formula
(4.7)f(x,y)-hf(x,y)=Dm,nf(ξ,η)m!n!ωxm,yn(x,y),(x,y)∈I2,(ξ,η)∈I2.
From the above error representation formula, for each g∈Bm,n we have hg=g to conclude that for a fixed (m,n) interpolation grid Gm,n there exists a unique transfinite interpolant hf from Bm,n to a given function f∈Cm,n.In addition, the univariate concept of a Haar-Chebyshev system can be extended in the infinite dimensional linear space of two-variable blending functions and interpolation grids consisting of vertical and horizontal line-segments as follows:ifh∈Bm,nsatisfiesh∣Gm,n=0,  thenh=0onI2. In other words, an(m,n)blending grid cannot be a zero set ofh∈Bm,n,hbeing nonidentically zero onI2.


## 4.4. Haar-Chebyshev Pair of a Linear Space and a Set of Grids in the Multivariate Case

In the one-dimensional Lagrange interpolation problem we have a finite set of points which are similar in the sense they obey the natural ordering on the real line. In the bivariate case we have set of grid-lines which are also similar to each other in the sense thatm of them are parallel to the one axis, and n of them, to the other. The points where the lines intersect the coordinate axes are also naturally ordered. Each blending grid Gm,n is unisolvent in transfinite Lagrange interpolation sense for Bm,n. On the other hand, the interpolation sets in the bivariate (multivariate) case may have diverse geometry and we can not talk about Haar-Chebyshev systems in general. However, we can argue that the pair of the linear space Bm,n and the set of blending grids 𝒢m,n(4.8){Bm,n,𝒢m,n}isaHaar-ChebyshevpaironI2,
where  𝒢m,n is the set of all blending grids of order (m,n) in I2, consisting of m vertical and n horizontal line-segments.


## 4.5. Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions

Let(4.9)Gm,n:={(x,y)∈I2:ωxm,yn(x,y)=0}
be (m,n) interpolation grid associated with the point sets xm:={xμ,1≤μ≤m} and yn:={yν,1≤ν≤n}. The interpolation grid Gm,n consists of m vertical and n horizontal line segments on I2. We associate with each point xμ a multiplicity sμ and with each point yν a multiplicity jν. Let ∑μ=1msμ=M and  ∑ν=1njν=N. Then for a given function f∈CM,N, there exists a unique transfinite Lagrange-Hermite blending interpolant hf from BM,N to f, satisfying the transfinite interpolation conditions
(4.10)Ds,0hf(xμ,y)=Ds,0f(xμ,y),s=0,…,sμ-1,μ=1,…,m,y∈[-1,1],D0,jhf(x,yν)=D0,jf(x,yν),j=0,…,jν-1,ν=1,…,n,x∈[-1,1]
with respect to the given (m,n) blending grid Gm,n, where as usual D0,0f=f.Explicit construction of the transfinite Lagrange-Hermite interpolanthf to f from BM,N is
(4.11)hf(x,y)=∑μ=1m∑s=0sμ-1Ds,0f(xμ,y)lμ,Ms(x)+∑ν=1n∑j=0jν-1D0,jf(x,yν)lν,Nj(y)-∑μ=1m∑s=0sμ-1∑ν=1n∑j=0jν-1Ds,jf(xμ,yν)lμ,Ms(x)lν,Nj(y),
where the fundamental Lagrange-Hermite interpolation polynomials lμ,Ms(x),μ=1,…,m and  lν,Nj(y),ν=1,…,n satisfy the interpolation conditions
(4.12)Dr,0lμ,Ms(xκ)=δμ,κδs,r,κ=1,…,m,r=0,…,sμ-1,D0,rlν,Nj(yκ)=δν,κδj,r,κ=1,…,n,r=0,…,jν-1.


## 4.6. Error Representation of Transfinite Lagrange-Hermite Interpolation by Algebraic Blending Functions in Terms of B-Splines

Let(4.13)Ω(xm,sm);(yn,jn)(x,y):=∏μ=1m(x-xμ)sμ∏ν=1n(y-yν)jν,
where sm:=(s1,…,sm) and jn:=(j1,…,jn). Then the following error representation formula holds
(4.14)f(x,y)-hf(x,y)=Ω(xm,sm);(yn,jn)(x,y)×∬-11BM-1;[(xm,sm),x](v)BN-1;[(yn,jn),x](w)DM,Nf(v,w)dvdw,
where BM-1;[(xm,sm),x] is the normalized B-spline of degree M-1 with knots (xm,x) of multiplicities (sm,1) and BN-1;[(yn,jn),y] is the normalized B-spline of degree N-1 with knots (yn,y) of multiplicities (jn,1). For details on B-splines see [11–13]. As a corollary we obtain Cauchy error representation for transfinite Lagrange-Hermite interpolation by blending functions
(4.15)f(x,y)-hf(x,y)=DM,Nf(ξ,η)M!N!Ω(xm,sm);(yn,jn)(x,y),(x,y)∈I2,(ξ,η)∈I2.
Note that hg=g for a blending function g∈BM,N. Therefore, each blending function from BM,N can be represented as transfinite Lagrange-Hermite interpolant on a grid consisting of horizontal and vertical line-segments.Remark 4.1.
The multivariate results which resemble the broadest extent the univariate theory concern transfinite interpolation and bestL1-approximation by algebraic blending functions. For simplicity of the notations the results on transfinite interpolation and best L1-approximation by algebraic blending functions are stated in the bivariate case d=2 although they are entirely valid in higher dimensions.


## 5. BestL1-Approximation by Algebraic Blending Functions

Let(ℱ,∥∘∥) be a normed linear space. Let U be a subspace of ℱ. Then uf∈U is called best approximant to f∈ℱ from U if for all u∈U(5.1)∥f-uf∥≤∥f-u∥.
Let f satisfy  Dm,nf(x,y)>0 on I2 and let h∈Bm,n. Then an (m,n) blending grid is a maximal set of zeros for f-h in a sense that if f(x,y)-h(x,y)=0 on (m,n) blending grid Gm,n, then f(x,y)-h(x,y)≠0 for (x,y)∈I2∖Gm,n. In other words, f-h cannot vanish on Gm,n∪(x0,y0), where (x0,y0)∈I2∖Gm,n.Approximation by blending functions is useful in problems of improving the efficiency of data transfer systems, image processing, reducing the size of the table of a function of many variables, cubature formulae, and numerical solution of differential and integral equations. For results on existence of best algebraic blendingL1-approximants see [14, 15]. The above notion of a Haar-Chebyshev system in the multivariate case is essential in the proof of the next theorem (see [16] for details).Theorem 5.1.
Letf∈Cm,n satisfy Dm,nf(x,y)≥0 for (x,y)∈I2. Then f possesses a unique best L1-approximant from Bm,n that is the unique transfinite Lagrange interpolant hf to f from Bm,n with respect to the blending grid
(5.2)G¯m,n={(x,y):∏i=1m∏j=1n(x-xi)(y-yj)=0},
where xk,1≤k≤m and yj,1≤j≤n are the zeros of the Chebyshev polynomials of second  kind Um(x) and Un(y), respectively.Remark 5.2.
Following Theorem5.1, we conclude that  the blending grid G¯m,n is the canonical set of best L1-approximation from Bm,n to the convex cone
(5.3){f∈Cm,n(I2):Dm,nf(x,y)≥0for(x,y)∈I2}.
Note that in the above convex cone, the non linear problem of best L1-approximation becomes a linear one.
Corollary by Theorem5.1 is the interesting fact that the best L1-approximant to the polynomial xmyn from Bm,n is also a polynomial, namely,
(5.4)∥Um(x)Un(y)2m+n∥L1(I2)≤∥xmyn-h(x,y)∥L1(I2),h∈Bm,n
and in particular the unique polynomial of minimal L1-norm (minimal L1-deviation) on I2 from the class of monic polynomials {∑i=0m∑j=0naijxiyj:amn=1} is 2-m-nUm(x)Un(y).Remark 5.3.
Note that in contrast to the result in Theorem5.1, the best uniform (Chebyshev) approximant from B1,1 to f∈C1,1 satisfying D1,1f(x,y)≥0 for (x,y)∈I2, is never unique unless f∈B1,1 (see [17] for details).


## 6. Lagrange-Hermite Transfinite Interpolation by Trigonometric Blending Functions

In bivariate case, transfinite interpolation by trigonometric blending functions is to construct a blending trigonometric interpolant over a planar domain in such a way that  matches (interpolates) a given function and its partial derivatives on curves that constitute the interpolation set. For details on the results in the present section see [7].LetC~2m+1,2n+1[0,2π)2 be the vector space of two-variable functions f(θ,η) which are 2π-periodic in each variable and
(6.1)𝔇(2m+1,2n+1)f(θ,η):=𝔇θ(2m+1)𝔇η(2n+1)f(θ,η)
is continuous on [0,2π)2, where 𝔇θ(2m+1)=𝔇θ∏k=1m(𝔇θ2+k2) and 𝔇θ=∂/∂θ is the partial derivative with respect to θ. The vector space Bm,nt of trigonometric blending functions of order(m,n) is defined as
(6.2)Bm,nt:={ht(θ,η)∈C~2m+1,2n+1[0,2π)2:𝔇(2m+1,2n+1)ht(θ,η)=0}.Eachtrigonometric blending functionht∈Bm,nt can be represented in the form
(6.3)ht(θ,η)=∑k=0m[ak(η)cos(kθ)+bk(η)sin(kθ)]+∑p=0n[cp(θ)cos(pη)+dp(θ)sin(pη)],
where ak,bk,cp, and dp are sufficiently smooth 2π-periodic functions.Remark 6.1.
Note that the above representation ofht∈Bm,nt is not unique in contrast to the uniqueness of the corresponding representation in Tn.
Letθ¯l1:=(θ0,θ1,…,θl1),0≤θ0<θ1<⋯<θl1<2π  be  l1+1 interpolation nodes with positive integer multiplicities λ¯l1:=(λ0,λ1,…,λl1) and let  η¯l2:=(η0,η1,…,ηl2),0≤η0<η1<⋯<ηl2<2π be l2+1 interpolation nodes with positive integer multiplicities β¯l2:=(β0,β1,…,βl2).


### 6.1. Construction of Lagrange-Hermite Transfinite Interpolation by Trigonometric Blending Functions

Letf(θ,η)∈C~2m+1,2n+1[0,2π)2. Given the interpolation nodes with corresponding multiplicities (θ¯l1,λ¯l1) and (η¯l2,β¯l2), ∑i=0l1λi=2m+1 and∑j=0l2βj=2n+1. Find a trigonometric blending function ht,f∈Bm,nt of order (m,n), satisfying the following Lagrange-Hermite interpolation conditions (f(i,j) denoting the partial derivative  ∂i+jf/∂θi∂ηj):
(6.4)ht,f(i,0)(θs1,η)=f(i,0)(θs1,η),i=0,1,…,λs1-1;s1=0,1,…,l1;ht,f(0,j)(θ,ηs2)=f(0,j)(θ,ηs2),j=0,1,…,βs2-1;s2=0,1,…,l2.Theorem 6.2.
Letf(θ,η)∈C~2m+1,2n+1[0,2π)2. Given the interpolation nodes with corresponding multiplicities (θ¯l1,λ¯l1) and (η¯l2,β¯l2) such that  ∑i=0l1λi=2m+1 and  ∑j=0l2βj=2n+1, then there exists a unique trigonometric blending interpolant ht,f from Bm,nt (of order (m,n)) to f satisfying the transfinite interpolation conditions (6.4). In addition, the following point-wise error representation holds:
(6.5)f(θ,η)-ht,f(θ,η)=𝔇(2m+1,2n+1)f(ξ,ζ)[m!]2[n!]2[B1(θ,η)-ht,B1(θ,η)],
where (ξ(θ,η),ζ(θ,η))∈(0,2π)2, B1(θ,η):=B1(θ)B1(η) and B1(θ)=θ-π,θ∈[0,2π) is the first univariate Bernoulli function.


### 6.2. Uniqueness of the Lagrange-Hermite Trigonometric Blending Interpolant

Suppose that there is another trigonometric blending interpolanth~t,f∈Bm,nt to f satisfying the interpolation conditions (6.4). Evidently,  ht,h~t,f=ht,f. Therefore, by using (6.5)
(6.6)h~t,f-ht,h~t,f=𝔇(2m+1,2n+1)h~t,f(ξ,ζ)[m!]2[n!]2[B1(θ,η)-ht,B1(θ,η)]=0⇒h~t,f=ht,h~t,f=ht,f.Lemma 6.3.
Letht,B1(θ,η) be the unique trigonometric blending interpolant from Bm,nt to B1(θ,η)=B1(θ)B1(η) satisfying the transfinite interpolation conditions (6.4) on the interpolating grid(6.7)Gl1,l2:={(θ,η)∈[0,2π]2:∏s1=0,s2=0l1,l2(θ-θs1)(η-ηs2)=0}.
Then,  [B1(θ,η)-ht,B1(θ,η)]≠0 for each point  (θ,η)∉Gl1,l2.Corollary 6.4.
Letht,f be the unique trigonometric blending interpolant from Bm,nt to f∈C~2m+1,2n+1[0,2π)2 satisfying the transfinite interpolation conditions (6.4). If 𝔇(2m+1,2n+1)f(ξ,ζ)≠0 for (ξ,ζ)∈[0,2π)2, then f(θ,η)-ht,f(θ,η)≠0 for (θ,η)∉Gl1,l2.


## 6.1. Construction of Lagrange-Hermite Transfinite Interpolation by Trigonometric Blending Functions

Letf(θ,η)∈C~2m+1,2n+1[0,2π)2. Given the interpolation nodes with corresponding multiplicities (θ¯l1,λ¯l1) and (η¯l2,β¯l2), ∑i=0l1λi=2m+1 and∑j=0l2βj=2n+1. Find a trigonometric blending function ht,f∈Bm,nt of order (m,n), satisfying the following Lagrange-Hermite interpolation conditions (f(i,j) denoting the partial derivative  ∂i+jf/∂θi∂ηj):
(6.4)ht,f(i,0)(θs1,η)=f(i,0)(θs1,η),i=0,1,…,λs1-1;s1=0,1,…,l1;ht,f(0,j)(θ,ηs2)=f(0,j)(θ,ηs2),j=0,1,…,βs2-1;s2=0,1,…,l2.Theorem 6.2.
Letf(θ,η)∈C~2m+1,2n+1[0,2π)2. Given the interpolation nodes with corresponding multiplicities (θ¯l1,λ¯l1) and (η¯l2,β¯l2) such that  ∑i=0l1λi=2m+1 and  ∑j=0l2βj=2n+1, then there exists a unique trigonometric blending interpolant ht,f from Bm,nt (of order (m,n)) to f satisfying the transfinite interpolation conditions (6.4). In addition, the following point-wise error representation holds:
(6.5)f(θ,η)-ht,f(θ,η)=𝔇(2m+1,2n+1)f(ξ,ζ)[m!]2[n!]2[B1(θ,η)-ht,B1(θ,η)],
where (ξ(θ,η),ζ(θ,η))∈(0,2π)2, B1(θ,η):=B1(θ)B1(η) and B1(θ)=θ-π,θ∈[0,2π) is the first univariate Bernoulli function.


## 6.2. Uniqueness of the Lagrange-Hermite Trigonometric Blending Interpolant

Suppose that there is another trigonometric blending interpolanth~t,f∈Bm,nt to f satisfying the interpolation conditions (6.4). Evidently,  ht,h~t,f=ht,f. Therefore, by using (6.5)
(6.6)h~t,f-ht,h~t,f=𝔇(2m+1,2n+1)h~t,f(ξ,ζ)[m!]2[n!]2[B1(θ,η)-ht,B1(θ,η)]=0⇒h~t,f=ht,h~t,f=ht,f.Lemma 6.3.
Letht,B1(θ,η) be the unique trigonometric blending interpolant from Bm,nt to B1(θ,η)=B1(θ)B1(η) satisfying the transfinite interpolation conditions (6.4) on the interpolating grid(6.7)Gl1,l2:={(θ,η)∈[0,2π]2:∏s1=0,s2=0l1,l2(θ-θs1)(η-ηs2)=0}.
Then,  [B1(θ,η)-ht,B1(θ,η)]≠0 for each point  (θ,η)∉Gl1,l2.Corollary 6.4.
Letht,f be the unique trigonometric blending interpolant from Bm,nt to f∈C~2m+1,2n+1[0,2π)2 satisfying the transfinite interpolation conditions (6.4). If 𝔇(2m+1,2n+1)f(ξ,ζ)≠0 for (ξ,ζ)∈[0,2π)2, then f(θ,η)-ht,f(θ,η)≠0 for (θ,η)∉Gl1,l2.


## 7. BestL1-Approximation by Trigonometric Blending Functions

Consider the normed vector spaceL1[0,2π]2 of functions that are 2π-periodic in each variable, whose absolute value has a finite integral on [0,2π]2 and equipped with the norm
(7.1)∥f∥L1[0,2π]2:=∫‍∫[0,2π]2|f(θ,η)|dA=(Fubini’sTheorem)∬02π|f(θ,η)|dθdη,
where A is the area measure. Denote for simplicity ∥f∥1:=∥f∥L1[0,2π]2. We restrict Bm,nt to Bm,nt⊂L1[0,2π]2.Let us consider the convex cone(7.2)C~+2m+1,2n+1:={f∈C~2m+1,2n+1[0,2π)2:𝔇(2m+1,2n+1)f(θ,η)≥0for(θ,η)∈[0,2π)2}.
We construct best L1-approximants from Bm,nt to functions in the convex cone C~+2m+1,2n+1. Following the interpolation problem (6.4) let θ¯={iπ/(m+1),i=1,…,2m+1} and η¯={jπ/(n+1),j=1,…,2n+1}, where all interpolation nodes are of multiplicity 1. By Theorem 6.2, there exists a unique trigonometric blending interpolant ht,f∈Bm,nt to f∈C~+2m+1,2n+1 on the blending grid
(7.3)G~m,n:={(θ,η)∈[0,2π]2:∏i=1,j=12m+1,2n+1[θ-iπ(m+1)][η-jπ(n+1)]=0},
consisting of 2m+1 vertical and 2n+1 horizontal line segments. The next theorem (see [7] for details) demonstrates that the interpolation grid G~m,n is the canonical sets of best L1-approximation from Bm,nt to the convex cone C~+2m+1,2n+1.Theorem 7.1.
The unique trigonometric blending interpolantht,f∈Bm,nt to f∈C~+2m+1,2n+1 on the blending grid G~m,n is the unique best L1-approximant to f from Bm,nt.


## 8. Canonical Set of BestL1-Approximation on a Triangle

Another canonical set result is presented in [18], where the domain is a triangle Δ∈R2. The approximating space consists of all functions that are sums of functions of the barycentric coordinates {λi}i=13 with respect to the vertices of Δ:
(8.1)B(Δ):={h∈C3(Δ):h(λ1,λ2)=h1(λ1)+h2(λ2)+h3(λ3),λ1+λ2+λ3=1,λi≥0}.B(Δ) is the kernel of the differential operator ∂3 which acts on the linear space C3(Δ) and is defined by
(8.2)∂3f(λ1,λ2):=∂∂λ1∂∂λ2(∂∂λ1-∂∂λ2)f(λ1,λ2).Letℳ:=ℳ1∪ℳ2∪ℳ3 be the union of the medians {ℳi}i=13 in Δ. The following interpolation theorem holds true.Theorem 8.1.
Letf∈C3(Δ). Then there exists a unique transfinite interpolant hf∈B(Δ) to f such that
(8.3)(hf)∣ℳ=f∣ℳ.
Moreover, if f∈Cm(Δ) for some m>3, then hf∈Cm(Δ) as well.On the basis of Theorem8.1 and an appropriate error representation formula, the corresponding best L1(Δ)-approximation is characterized in terms of a canonical point set. The convex cone here is defined by another differential operator 𝒟~ (see [18, page 456] for details) that is represented in terms of ∂3 in a bit intricate way involving infinite series. The canonical result of best L1-approximation reads as follows.Theorem 8.2.
Letf∈C3(Δ) satisfy 𝒟~f≥0 on Δ. Then, the unique transfinite interpolant hf from Theorem 8.1 is the unique best L1-approximant to the function f from B(Δ).Remark 8.3.
By Theorem8.2, the point set ℳ is a canonical set of best L1-approximation on a triangle.


## 9. BestL1-Approximation by Harmonic Functions

Here, we discuss results on bestL1-approximation of subharmonic functions by harmonic functions. Let B(r) denote the open ball centered at 0 of radius r>0 in the d-dimensional space Rd and let B:=B(1) be the unit ball centered at 0 in Rd and B0:=B(2-1/d). Denote by ℋ(B) the linear space of harmonic functions on B and let 𝒮(B) denote the convex cone of subharmonic functions on B. Let f∈C(B¯) be continuous on the closed unit ball and let f∈𝒮(B). A function hf∈C(B¯)∩ℋ(B) is called a best harmonic L1-approximant to f on B if
(9.1)∥f-hf∥L1(B)≤∥f-h∥L1(B)
for all h∈C(B¯)∩ℋ(B). The next theorem gives a canonical set characterization of the best harmonic L1-approximants to the convex cone C(B¯)∩𝒮(B) (see [19] for details).Theorem 9.1.
Letf∈C(B¯)∩𝒮(B). Then the harmonic function hf is a best harmonic L1-approximant from C(B¯)∩ℋ(B)  to  f  if and only if (i)
hf=f on ∂B0 (that is,  hf is a transfinite harmonic interpolant to  f on  ∂B0);(ii)
hf≤fonB¯∖B0.Remark 9.2.
If a best harmonicL1-approximant to f∈C(B¯)∩𝒮(B) exists, then it is unique. However, a best harmonic L1-approximant to an arbitrary function f∈C(B¯)∩𝒮(B) need not exist. For example, g(x,y):=x4y4 is a subharmonic polynomial on the unit disk in R2 which does not possess a best L1-approximant from C(B¯)∩ℋ(B). However obviously, D2,2g(x,y)≥0 and according to Theorem 5.1, a unique best L1-approximant on [-1,1]2 to g from B2,2 exists. For other results on best harmonic L1-approximation see [20, 21].Remark 9.3.
The constructive characterization of Theorem9.1 is in terms of the Dirichlet problem on ∂B0. It can be considered as a transfinite interpolation problem by harmonic functions on the spherical interpolation grid ∂B0. The canonical set of best harmonic L1-approximation to the convex cone C(B¯)∩𝒮(B) on the unit ball is ∂B0.


## 10. Best One-SidedL1-Approximation by Algebraic Blending Functions

Letf∈C(I2). A function h*∈Bm,n is called best one-sided from above L1-approximant to f from the linear space Bm,n if
(10.1)h*(x,y)≥f(x,y),(x,y)∈I2,∫I2(h*-f)≤∫I2(h-f)
for all h∈Bm,n(I2) such that h(x,y)≥f(x,y),(x,y)∈I2. Analogously, we define best one-sided from below L1-approximant to f from Bm,n.This type of one-sidedL1-approximation (with respect to a convex set rather than a subspace) has been a subject of much research activity (see [1] for details). The results in this area have mainly dealt with the case when the best approximant is from a finite-dimensional linear space. However, as we have mentioned above, the linear space Bm,n of algebraic blending functions of order (m,n) is of infinite dimension.First we reformulate a general result in the particular case ofBm,n approximating linear space (see [22, 23] for details). It shows that the canonical sets must satisfy certain conditions.Theorem 10.1.
Letf∈C(I2). Let
(10.2)𝒰(f):={h∈Bm,n:h(x,y)≥f(x,y)onI2}.
Let h*∈𝒰(f) and let Z:=Z(h*-f) be the zero set of (h*-f) in I2. Then, the following are equivalent.(a)
The blending functionh* is a best one-sided from above L1-approximant to f from Bm,n on I2.(b)
Gaussian property for the zero setZ holds
(10.3)∫I2h≥0wheneverh∈Bm,n,h≥0onZ.(c)
The domainI2 is a quadrature domain with respect to the point set Z and the linear space Bm,n;  that is, there is a positive measure μ with support in Z such that
(10.4)∫I2h=∫Zhdμ
for all h∈Bm,n.By analogy with the canonical sets in the bestL1-approximation by blending functions one can expect that the best one-sided L1-approximants by algebraic blending functions are Lagrange-Hermite transfinite interpolants on grids consisting of vertical and horizontal lines. However, Theorem 10.1 and the next result (see [24] for details) show that this is not the case: transfinite interpolation grids consisting of vertical and horizontal line-segments inI2cannot be canonical sets of best one-sidedL1-approximation by blending functions.Theorem 10.2.
Letm≥1 and n≥1. Then there exists an algebraic blending function  h∈B2m,2n such that it is positive on the Legendre (m,n) grid
(10.5)Gm,n,L:={(x,y)∈I2:Lm(x)Ln(y)=0},
where Lm and Ln are the Legendre polynomials of degree m, respectivelyn, that is,
(10.6)h∣Gm,n,L>0,
but its integral on I2 is negative:
(10.7)∫I2h<0.Analogous result concerning best approximation by trigonometric blending functions is published in [7].


### 10.1. Best One-SidedL1-Approximation by Algebraic B1,1-Blending Functions

We give a constructive characterization in terms of canonical sets for the best one-sidedL1-approximant to a function f∈C1,1 satisfying D1,1f≥0 on I2 from the infinite-dimensional linear space B1,1. The best L1-approximants are transfinite Lagrange-Hermite interpolants on the diagonals of I2 as canonical sets. The occurrence of the diagonals as canonical sets of best one-sidedL1-approximation (not interpolation grids consisting of vertical and horizontal line-segments inI2!) is a fact which, to the best of our knowledge, has been first observed in [25].Note that(10.8)B1,1={h(x,y):h(x,y)=a(x)+b(y),a∈C1(I),b∈C1(I)}.
In other words, we  approximate two-variable functions by sums of univariate functions on I2.The proof of existence, uniqueness and explicit construction of the best one-sidedL1-approximant from above and from below consists of three main steps (see [25, 26] for details).(A) Transfinite Lagrange-Hermite Interpolation Formula on an Appropriate Grid with an Error Remainder TermTheorem 10.3.
Letf∈C1,1.
(a) TheB1,1 blending function
(10.9)hf*(x,y):=f(-1,-1)+∫-1xD1,0f(t,t)dt+∫-1yD0,1f(t,t)dt
is the unique Lagrange-Hermite transfinite interpolant to f from B1,1 satisfying the transfinite interpolation conditions
(10.10)(hf*)∣Δ*=f∣Δ*,(gradhf*)∣Δ*=(gradf)∣Δ*
on  the main diagonal  Δ* of I2. Moreover, the error representation formula holds
(10.11)f(x,y)-hf*(x,y)=-D1,1f(ξ,η)2(x-y)2,(x,y)∈I2,(ξ,η)∈I2.
(b) TheB1,1 blending function
(10.12)h*f(x,y):=f(-1,1)+∫-1xD1,0f(t,-t)dt-∫-1-yD0,1f(t,-t)dt
is the unique Lagrange-Hermite transfinite interpolant to f from B1,1 satisfying the transfinite interpolation conditions
(10.13)(h*f)∣Δ*=f∣Δ*,(grad)h*f∣Δ*=(grad)f∣Δ*
on the antidiagonal Δ* of I2. Moreover, the following error representation formula holds:
(10.14)f(x,y)-h*f(x,y)=D1,1f(ρ,σ)2(x+y)2,(x,y)∈I2,(ρ,σ)∈I2.The following observation is an essential fact in the proof of the next theorem. Letf satisfy D1,1f(x,y)>0 on I2 and let h∈B1,1(I2). If (f-h)=0 on Δ* and (grad(f-h))|Δ*=(0,0), then  (f-h)≠0 on I2∖Δ*. In other words, (f-h) can not vanish on   Δ*∪(x0,y0), where (x0,y0)∈I2∖Δ*.(B) Transfinite Cubature Formulae on
Δ
*
and
Δ
*
with
B
1,1
Blending Degree of PrecisionLetf∈C1,1. Then
(10.15)∫I2f-2∫If(x,x)dx=-43D1,1f(α,β),(α,β)∈I2.
In particular
(10.16)∫I2h=2∫Ih(x,x)dx,h∈B1,1.
Analogously,
(10.17)∫I2f-2∫I(x,-x)dx=43D1,1f(γ,δ),(γ,δ)∈I2.
In particular
(10.18)∫I2h=2∫Ih(x,-x)dx,h∈B1,1.(C) Constructive Characterization of the Best One-Sided
L
1
-Approximants from
B
1,1.Theorem 10.4.
Letf∈C1,1 satisfy D1,1f(x,y)≥0 for (x,y)∈I2. Then we have the following.(a)
The functionf has a unique best one-sided from above L1-approximantfrom  B1,1. The unique best one-sided from above L1-approximant  hf*  is  characterized by  the Lagrange-Hermite type transfinite interpolation conditions on the main diagonal Δ* of  I2:
(10.19)(hf*)|Δ*=f|Δ*,(gradhf*)∣Δ*=(gradf)∣Δ*.(b)
The functionf has a unique best one-sided from below L1-approximant from B1,1. Theunique best one-sided from below L1-approximant h*f is characterized by  the Lagrange-Hermite type transfinite interpolation conditions on the antidiagonal Δ*  of I2:
(10.20)(h*f)∣Δ*=f∣Δ*,(gradh*f)∣Δ*=(gradf)∣Δ*.Proof of Theorem10.4.
We sketch the proof of  (a). The proof of (b) follows the same steps. Givenf∈C1,1 such that D1,1f(x,y)≥0 for (x,y)∈I2. By Theorem 10.3, hf*≥f on I2. Consider an arbitrary blending function h∈B1,1 such that h≥f on I2. By using the transfinite cubature formula from (B), we have
(10.21)∫I2(h-f)=∫I2(h-hf*)+∫I2(hf*-f)=2∫-11(h-hf*)(x,x)dx+∫I2(hf*-f)≥∫I2(hf*-f),
taking into account that ∫-11(h-h*)(x,x)dx≥0 which follows from h≥f on I2 and (hf*)∣Δ*=f∣Δ*. Hence,
(10.22)∫I2‍(h-f)≥∫I2‍(hf*-f),
for  each function h∈B1,1 satisfying h≥f on I2.Uniqueness of the Best One-Sided from above
L
1
-Approximant from
B
1,1
to
f
in the Convex Cone
{
f
∈
C
1,1
:
D
1,1
f
(
x
,
y
)
≥
0
,
(
x
,
y
)
∈
I
2
}
.Suppose thath∈B1,1 is another best one-sided L1-approximant to f from B1,1. It follows from (10.21)  that  f∣Δ*=(hf*)∣Δ*=h∣Δ*. From h≥f on I2 we conclude that (gradh)∣Δ*=(gradf)∣Δ*.  By Theorem 10.3, h=hf*.  The proof is completed.Example 10.5.
Considerf(x,y)=xy,D1,1f(x,y)=1>0. According to Theorem 10.4, the transfinite Lagrange-Hermite interpolant to f from B1,1(10.23)hf*(x,y)=f(-1,-1)+∫-1xD1,0f(t,t)dt+∫-1yD0,1f(t,t)dt=1+∫-1xtdt+∫-1ytdt=x2+y22
is the unique best one-sided from above L1-approximant from B1,1 to f(x,y)=xy (see Figure 1).Figure 1
f
(
x
,
y
)
=
x
y   and   hf*(x,y).Example 10.6.
Letf(x,y)=xpyq,  x≥0,p>0,y≥0,q>0,  p+q=1. We compute D1,0f(x,y)=pxp-1yq,  D0,1f(x,y)=qxpyq-1,  D1,1f(x,y)=pqxp-1yq-1>0 for x,y>0. According to Theorem 10.4, the transfinite Lagrange-Hermite interpolant to f from B1,1(10.24)hf*(x,y)=f(0,0)+∫0xD1,0f(t,t)dt+∫0yD0,1f(t,t)dt=∫0xpdt+∫0yqdt=px+qy
is the unique best one-sided from above L1-approximant from B1,1 to  f(x,y)=xpyq on [0,1]2.


### 10.2. Best One-SidedL1-Approximation by Algebraic B2,1-Blending Functions

Next theorem (see [27] for details) gives characterization of the best one-sided L1-approximants by algebraic B2,1-blending functions in terms of transfinite Lagrange-Hermite interpolation on canonical sets.Theorem 10.7.
Letf∈C2,1  and  D2,1f(x,y)≥0 for (x,y)∈I2.
(a) Let(10.25)v*:={(x,2|x|-1):x∈[-1,1]}.
The function f possesses a unique best one-sided from above L1-approximant h* from B2,1. The unique best one-sided from above L1-approximant hf* to f is characterized by the simultaneous Lagrange-Hermite type transfinite interpolation conditions:
(10.26)(hf*)∣v*=f∣v*,(gradhf*)∣v*=(gradf)∣v*.
(b) Let(10.27)v*:={(x,1-2|x|):x∈[-1,1]}.
The function f possesses a unique best one-sided from below L1-approximant h*f from B2,1. The unique best one-sided from below L1-approximant h*f to f is characterized by the simultaneous Lagrange-Hermite type transfinite interpolation condition:
(10.28)(h*f)∣v*=f∣v*,(gradh*f)∣v*=(gradf)∣v*.The following two steps (see [27] for details) are essential in the proof of Theorem 10.7.(A) Transfinite Interpolation Formulas with Remainder Term
Letf∈C2,1 and let   hf*  and  h*f be the transfinite interpolants to f from Theorem 5.1. Then
(10.29)f(x,y)-hf*(x,y)=-D2,1f(ξ1,η1)24(y-2|x|+1)2(y+4|x|+1),
where (x,y)∈I2 and  (ξ1,η1)∈I2 and
(10.30)f(x,y)-h*f(x,y)=D2,1f(ξ2,η2)24(y-2|x|+1)2(-y+4|x|+1),
where (x,y)∈I2 and (ξ2,η2)∈I2.(B) Transfinite Cubature Formulae onv* and v* with B2,1 Blending Degree of Precision
Letf∈C2,1. Then
(10.31)∫I2f=2∫-11f(x,2|x|-1)dx-13D2,1f(ρ1,σ1),(ρ1,σ1)∈I2,∫I2f=2∫-11f(x,1-2|x|)dx+13D2,1f(ρ2,σ2),(ρ2,σ2)∈I2.Example 10.8.
Considerf(x,y)=x2y,   D2,1f(x,y)=2>0 on I2. Then, by Theorem 10.7, the B2,1 transfinite blending interpolant
(10.32)hf*(x,y):=43x2|x|-x2+112(1+y)3
to f is the unique best one-sided from above L1-approximant from B2,1 to f(x,y)=x2y. Analogously, the B2,1 transfinite interpolant
(10.33)h*f(x,y):=-43x2|x|+x2-112(1+y)3
to f is the unique best one-sided from below L1-approximant from B2,1 to f(x,y)=x2y (see Figure 2). Note that the best one-sided approximants to the polynomial f(x,y)=x2y are not polynomials. Moreover, they have a limited C2,1-smoothness, contrary to the unconstrained best L1-approximation by blending functions (see Theorem 5.1).Figure 2
f
(
x
,
y
)
=
x
2
y, hf*(x,y)  and  h*f(x,y).


### 10.3. Best One-SidedL1-Approximation by B2,2-Blending Functions

Letf∈C2,2 and D2,2f(x,y)≥0 for (x,y)∈I2. Then (see [28] for details), the unique best one-sided from above L1-approximant hf* to f from B2,2 is the unique transfinite Lagrange-Hermite interpolant to f on the canonical grid
(10.34)x:={(x,y)∈I2:|x|=|y|}
satisfying the transfinite interpolation conditions
(10.35)(hf*)|x=f∣x,(gradhf*)∣x=(gradf)∣x.
For f(x,y) even with respect to one of the variables x or y, the unique best one-sided from below L1-approximant h*f to f from B2,2 is the unique transfinite Lagrange-Hermite interpolant to f on the canonical grid
(10.36)⋄:={(x,y)∈I2:|x|+|y|=1}
satisfying the transfinite interpolation conditions
(10.37)(h*f)∣⋄=f∣⋄,(gradh*f)∣⋄=(gradf)∣⋄.
Surprisingly, there is no universal canonical grid for the entire convex cone
(10.38){f∈C2,2,D2,2f(x,y)≥0,(x,y)∈I2}
concerning the best one-sided from below L1-approximation from B2,2 (see [28] for details).The best one-sided from aboveL1-approximant to f has the smoothness of f. The best one-sided from below L1-approximant to f (if it exists) is a blending transfinite spline function with two line-segment knots{(x,0):x∈I} and {(0,y):y∈I}.Example 10.9.
Considerf(x,y)=x2y2, D2,2f(x,y)=4>0. Then hf*(x,y):=(x4+y4)/2 is the unique best one-sided from above L1-approximant to f(x,y)=x2y2 from B2,2 and
(10.39)h*f(x,y):=12(x4+y4)-43(|x|3+|y|3)+x2+y2-16
is the unique best one-sided from below L1-approximant to f(x,y)=x2y2 from B2,2 (see Figure 3). Note that h*∉C3,0∪C0,3. Hence, the best one-sided from below L1-approximant from B2,2 to f does not inherit the smoothness of f.Figure 3
f
(
x
,
y
)
=
x
2
y
2, hf*(x,y)  and  h*f(x,y).


## 10.1. Best One-SidedL1-Approximation by Algebraic B1,1-Blending Functions

We give a constructive characterization in terms of canonical sets for the best one-sidedL1-approximant to a function f∈C1,1 satisfying D1,1f≥0 on I2 from the infinite-dimensional linear space B1,1. The best L1-approximants are transfinite Lagrange-Hermite interpolants on the diagonals of I2 as canonical sets. The occurrence of the diagonals as canonical sets of best one-sidedL1-approximation (not interpolation grids consisting of vertical and horizontal line-segments inI2!) is a fact which, to the best of our knowledge, has been first observed in [25].Note that(10.8)B1,1={h(x,y):h(x,y)=a(x)+b(y),a∈C1(I),b∈C1(I)}.
In other words, we  approximate two-variable functions by sums of univariate functions on I2.The proof of existence, uniqueness and explicit construction of the best one-sidedL1-approximant from above and from below consists of three main steps (see [25, 26] for details).(A) Transfinite Lagrange-Hermite Interpolation Formula on an Appropriate Grid with an Error Remainder TermTheorem 10.3.
Letf∈C1,1.
(a) TheB1,1 blending function
(10.9)hf*(x,y):=f(-1,-1)+∫-1xD1,0f(t,t)dt+∫-1yD0,1f(t,t)dt
is the unique Lagrange-Hermite transfinite interpolant to f from B1,1 satisfying the transfinite interpolation conditions
(10.10)(hf*)∣Δ*=f∣Δ*,(gradhf*)∣Δ*=(gradf)∣Δ*
on  the main diagonal  Δ* of I2. Moreover, the error representation formula holds
(10.11)f(x,y)-hf*(x,y)=-D1,1f(ξ,η)2(x-y)2,(x,y)∈I2,(ξ,η)∈I2.
(b) TheB1,1 blending function
(10.12)h*f(x,y):=f(-1,1)+∫-1xD1,0f(t,-t)dt-∫-1-yD0,1f(t,-t)dt
is the unique Lagrange-Hermite transfinite interpolant to f from B1,1 satisfying the transfinite interpolation conditions
(10.13)(h*f)∣Δ*=f∣Δ*,(grad)h*f∣Δ*=(grad)f∣Δ*
on the antidiagonal Δ* of I2. Moreover, the following error representation formula holds:
(10.14)f(x,y)-h*f(x,y)=D1,1f(ρ,σ)2(x+y)2,(x,y)∈I2,(ρ,σ)∈I2.The following observation is an essential fact in the proof of the next theorem. Letf satisfy D1,1f(x,y)>0 on I2 and let h∈B1,1(I2). If (f-h)=0 on Δ* and (grad(f-h))|Δ*=(0,0), then  (f-h)≠0 on I2∖Δ*. In other words, (f-h) can not vanish on   Δ*∪(x0,y0), where (x0,y0)∈I2∖Δ*.(B) Transfinite Cubature Formulae on
Δ
*
and
Δ
*
with
B
1,1
Blending Degree of PrecisionLetf∈C1,1. Then
(10.15)∫I2f-2∫If(x,x)dx=-43D1,1f(α,β),(α,β)∈I2.
In particular
(10.16)∫I2h=2∫Ih(x,x)dx,h∈B1,1.
Analogously,
(10.17)∫I2f-2∫I(x,-x)dx=43D1,1f(γ,δ),(γ,δ)∈I2.
In particular
(10.18)∫I2h=2∫Ih(x,-x)dx,h∈B1,1.(C) Constructive Characterization of the Best One-Sided
L
1
-Approximants from
B
1,1.Theorem 10.4.
Letf∈C1,1 satisfy D1,1f(x,y)≥0 for (x,y)∈I2. Then we have the following.(a)
The functionf has a unique best one-sided from above L1-approximantfrom  B1,1. The unique best one-sided from above L1-approximant  hf*  is  characterized by  the Lagrange-Hermite type transfinite interpolation conditions on the main diagonal Δ* of  I2:
(10.19)(hf*)|Δ*=f|Δ*,(gradhf*)∣Δ*=(gradf)∣Δ*.(b)
The functionf has a unique best one-sided from below L1-approximant from B1,1. Theunique best one-sided from below L1-approximant h*f is characterized by  the Lagrange-Hermite type transfinite interpolation conditions on the antidiagonal Δ*  of I2:
(10.20)(h*f)∣Δ*=f∣Δ*,(gradh*f)∣Δ*=(gradf)∣Δ*.Proof of Theorem10.4.
We sketch the proof of  (a). The proof of (b) follows the same steps. Givenf∈C1,1 such that D1,1f(x,y)≥0 for (x,y)∈I2. By Theorem 10.3, hf*≥f on I2. Consider an arbitrary blending function h∈B1,1 such that h≥f on I2. By using the transfinite cubature formula from (B), we have
(10.21)∫I2(h-f)=∫I2(h-hf*)+∫I2(hf*-f)=2∫-11(h-hf*)(x,x)dx+∫I2(hf*-f)≥∫I2(hf*-f),
taking into account that ∫-11(h-h*)(x,x)dx≥0 which follows from h≥f on I2 and (hf*)∣Δ*=f∣Δ*. Hence,
(10.22)∫I2‍(h-f)≥∫I2‍(hf*-f),
for  each function h∈B1,1 satisfying h≥f on I2.Uniqueness of the Best One-Sided from above
L
1
-Approximant from
B
1,1
to
f
in the Convex Cone
{
f
∈
C
1,1
:
D
1,1
f
(
x
,
y
)
≥
0
,
(
x
,
y
)
∈
I
2
}
.Suppose thath∈B1,1 is another best one-sided L1-approximant to f from B1,1. It follows from (10.21)  that  f∣Δ*=(hf*)∣Δ*=h∣Δ*. From h≥f on I2 we conclude that (gradh)∣Δ*=(gradf)∣Δ*.  By Theorem 10.3, h=hf*.  The proof is completed.Example 10.5.
Considerf(x,y)=xy,D1,1f(x,y)=1>0. According to Theorem 10.4, the transfinite Lagrange-Hermite interpolant to f from B1,1(10.23)hf*(x,y)=f(-1,-1)+∫-1xD1,0f(t,t)dt+∫-1yD0,1f(t,t)dt=1+∫-1xtdt+∫-1ytdt=x2+y22
is the unique best one-sided from above L1-approximant from B1,1 to f(x,y)=xy (see Figure 1).Figure 1
f
(
x
,
y
)
=
x
y   and   hf*(x,y).Example 10.6.
Letf(x,y)=xpyq,  x≥0,p>0,y≥0,q>0,  p+q=1. We compute D1,0f(x,y)=pxp-1yq,  D0,1f(x,y)=qxpyq-1,  D1,1f(x,y)=pqxp-1yq-1>0 for x,y>0. According to Theorem 10.4, the transfinite Lagrange-Hermite interpolant to f from B1,1(10.24)hf*(x,y)=f(0,0)+∫0xD1,0f(t,t)dt+∫0yD0,1f(t,t)dt=∫0xpdt+∫0yqdt=px+qy
is the unique best one-sided from above L1-approximant from B1,1 to  f(x,y)=xpyq on [0,1]2.


## 10.2. Best One-SidedL1-Approximation by Algebraic B2,1-Blending Functions

Next theorem (see [27] for details) gives characterization of the best one-sided L1-approximants by algebraic B2,1-blending functions in terms of transfinite Lagrange-Hermite interpolation on canonical sets.Theorem 10.7.
Letf∈C2,1  and  D2,1f(x,y)≥0 for (x,y)∈I2.
(a) Let(10.25)v*:={(x,2|x|-1):x∈[-1,1]}.
The function f possesses a unique best one-sided from above L1-approximant h* from B2,1. The unique best one-sided from above L1-approximant hf* to f is characterized by the simultaneous Lagrange-Hermite type transfinite interpolation conditions:
(10.26)(hf*)∣v*=f∣v*,(gradhf*)∣v*=(gradf)∣v*.
(b) Let(10.27)v*:={(x,1-2|x|):x∈[-1,1]}.
The function f possesses a unique best one-sided from below L1-approximant h*f from B2,1. The unique best one-sided from below L1-approximant h*f to f is characterized by the simultaneous Lagrange-Hermite type transfinite interpolation condition:
(10.28)(h*f)∣v*=f∣v*,(gradh*f)∣v*=(gradf)∣v*.The following two steps (see [27] for details) are essential in the proof of Theorem 10.7.(A) Transfinite Interpolation Formulas with Remainder Term
Letf∈C2,1 and let   hf*  and  h*f be the transfinite interpolants to f from Theorem 5.1. Then
(10.29)f(x,y)-hf*(x,y)=-D2,1f(ξ1,η1)24(y-2|x|+1)2(y+4|x|+1),
where (x,y)∈I2 and  (ξ1,η1)∈I2 and
(10.30)f(x,y)-h*f(x,y)=D2,1f(ξ2,η2)24(y-2|x|+1)2(-y+4|x|+1),
where (x,y)∈I2 and (ξ2,η2)∈I2.(B) Transfinite Cubature Formulae onv* and v* with B2,1 Blending Degree of Precision
Letf∈C2,1. Then
(10.31)∫I2f=2∫-11f(x,2|x|-1)dx-13D2,1f(ρ1,σ1),(ρ1,σ1)∈I2,∫I2f=2∫-11f(x,1-2|x|)dx+13D2,1f(ρ2,σ2),(ρ2,σ2)∈I2.Example 10.8.
Considerf(x,y)=x2y,   D2,1f(x,y)=2>0 on I2. Then, by Theorem 10.7, the B2,1 transfinite blending interpolant
(10.32)hf*(x,y):=43x2|x|-x2+112(1+y)3
to f is the unique best one-sided from above L1-approximant from B2,1 to f(x,y)=x2y. Analogously, the B2,1 transfinite interpolant
(10.33)h*f(x,y):=-43x2|x|+x2-112(1+y)3
to f is the unique best one-sided from below L1-approximant from B2,1 to f(x,y)=x2y (see Figure 2). Note that the best one-sided approximants to the polynomial f(x,y)=x2y are not polynomials. Moreover, they have a limited C2,1-smoothness, contrary to the unconstrained best L1-approximation by blending functions (see Theorem 5.1).Figure 2
f
(
x
,
y
)
=
x
2
y, hf*(x,y)  and  h*f(x,y).


## 10.3. Best One-SidedL1-Approximation by B2,2-Blending Functions

Letf∈C2,2 and D2,2f(x,y)≥0 for (x,y)∈I2. Then (see [28] for details), the unique best one-sided from above L1-approximant hf* to f from B2,2 is the unique transfinite Lagrange-Hermite interpolant to f on the canonical grid
(10.34)x:={(x,y)∈I2:|x|=|y|}
satisfying the transfinite interpolation conditions
(10.35)(hf*)|x=f∣x,(gradhf*)∣x=(gradf)∣x.
For f(x,y) even with respect to one of the variables x or y, the unique best one-sided from below L1-approximant h*f to f from B2,2 is the unique transfinite Lagrange-Hermite interpolant to f on the canonical grid
(10.36)⋄:={(x,y)∈I2:|x|+|y|=1}
satisfying the transfinite interpolation conditions
(10.37)(h*f)∣⋄=f∣⋄,(gradh*f)∣⋄=(gradf)∣⋄.
Surprisingly, there is no universal canonical grid for the entire convex cone
(10.38){f∈C2,2,D2,2f(x,y)≥0,(x,y)∈I2}
concerning the best one-sided from below L1-approximation from B2,2 (see [28] for details).The best one-sided from aboveL1-approximant to f has the smoothness of f. The best one-sided from below L1-approximant to f (if it exists) is a blending transfinite spline function with two line-segment knots{(x,0):x∈I} and {(0,y):y∈I}.Example 10.9.
Considerf(x,y)=x2y2, D2,2f(x,y)=4>0. Then hf*(x,y):=(x4+y4)/2 is the unique best one-sided from above L1-approximant to f(x,y)=x2y2 from B2,2 and
(10.39)h*f(x,y):=12(x4+y4)-43(|x|3+|y|3)+x2+y2-16
is the unique best one-sided from below L1-approximant to f(x,y)=x2y2 from B2,2 (see Figure 3). Note that h*∉C3,0∪C0,3. Hence, the best one-sided from below L1-approximant from B2,2 to f does not inherit the smoothness of f.Figure 3
f
(
x
,
y
)
=
x
2
y
2, hf*(x,y)  and  h*f(x,y).


## 11. Best One-SidedL1-Approximation by Quasi-Blending Functions

The characterization result of best one-sidedL1-approximation by algebraic B2,1-blending functions (see Theorem 10.7) has a natural extension to approximation by quasi-blending functions (see [29] for details). Define the space QBm,1 of all quasi-blending functions of order (m,1) by
(11.1)QBm,1:={h∈C0,1(I2):D0,1h(·,y)∈πm-1forafixedy∈I},
where πm-1 is the space of all univariate algebraic polynomials of degree not exceeding m-1. Any h∈QBm,1 can be represented in the form
(11.2)h(x,y)=b(x)+∑μ=0m-1aμ(y)xμ,(x,y)∈I2,
with a0,a1,…,am-1∈C1(I) and b∈C0(I). Obviously, this representation is not unique. The canonical sets v* and v* for the best L1-approximation by B2,1-blending functions (see Theorem 10.7) are examples of the so-called m-oscillating point sets.Givenm+1 points -1=x0<x1<⋯<xm=1. Let η∈{-1,1}. A continuous function φ:I→I is called m-oscillating if  (i)
φ(xμ)=η(-1)μ,  0≤μ≤m;(ii)
The restrictionφμ:=φ|[xμ-1,xμ] is a homeomorphism between [xμ-1,xμ] and I.A point setΓ∈I2 is called m-oscillating if it is the graph of an m-oscillating function. The m-oscillating point sets are unisolvent for the following transfinite interpolation by quasi-blending functions.Letm∈N, f∈Cm,1(I2), and Γ be an m-oscillating point set associated with a given m-oscillating function φ. Then, there exists a unique quasi-blending function h∈QBm,1 satisfying the following transfinite interpolation conditions:
(11.3)h∣Γ=f∣Γ,(D0,1h)∣Γ=(D0,1f)∣Γ.We define twom-oscillating point sets which turn out to be canonical sets of best L1-approximation by quasi-blending functions from QBm,1. Given m∈N and x,s∈(-1,1), consider the functions
(11.4)qm(x,s):={Pk((1+s)/2,-(1+s)/2)(x)Pk((1+s)/2,-(1+s)/2)(-x),ifm=2k,Pk(-(1-s)/2,-(1+s)/2)(x)Pk((1-s)/2,(1+s)/2)(x),ifm=2k+1,
where Pn(α,β) denote the corresponding Jacobi polynomials of degree n.  Each polynomial qm(·,s) has m distinct zeros -1<ψ1(s)<⋯<ψm(s)<1. Moreover, it can be shown (see [29] for details) that the inverse functions φμ:=ψμ-1:[xμ-1,xμ]→I,  1≤μ≤m exist, are continuously differentiable, and join together to an m-oscillating function φ:I→I such that the quadrature formulae
(11.5)∫I2f(x,y)dxdy≈2∫-11f(s,φ(s))ds;∫I2f(x,y)dxdy≈2∫I2f(s,-φ(s))ds
are exact in the space QBm,1. Let the m-oscillating point sets Γm and Lm be given by
(11.6)Γm:={(x,(-1)mφ(x))∈I2:x∈I},Lm:={(x,(-1)m+1φ(x))∈I2:x∈I}.
The next theorem shows that the sets Γm and Lm are canonical sets of best one-sided L1-approximation from QBm,1.Theorem 11.1.
Letf∈Cm,1(I2) satisfy Dm,1f≥0 on I2. Then the following holds. (a)
The function f possesses a unique best one-sidedL1-approximant from above from  QBm,1. The best one-sided from above approximant hf* is  characterized by the following Lagrange-Hermite transfinite interpolation conditions:
(11.7)(i)(hf*)∣Γm=f∣Γm,(ii)(D0,1hf*)∣Γm=(D0,1f)∣Γm.(b)
The unique best one-sided from belowL1-approximant h*f to f from QBm,1 is characterized by the following Lagrange-Hermite transfinite interpolation conditions:
(11.8)(i)(h*f)∣Lm=f∣Lm,(ii)(D0,1hf*)∣Lm=(D0,1f)∣Lm.


## 12. Best One-SidedL1-Approximation by Sums of Univariate Functions

Here we present a multivariate extension of Theorem10.4 (see [30] for details). Let d≥2 and C1,d be the linear space of all differentiable functions f, defined on the d-dimensional cube Id:=[-1,1]d and having continuous mixed derivatives
(12.1)Di,j1,1f:=∂2f∂xi∂xj,1≤i<j≤d.
We denote by B1,d the subspace of all d-variable functions h which are sums of univariate ones, that is,
(12.2)B1,d:={h∈C1,d:h(x)=h(x1,…,xd):=∑i=1dhi(xi)}.
or equivalently
(12.3)B1,d={h∈C1,d:Di,j1,1h=0,1≤i<j≤d}.
Let Δ*:={(t,…,t)∈Id:t∈[-1,1]} be the main diagonal of the d-dimensional cube Id. Let
(12.4)gradf:=(∂f∂x1,…,∂f∂xd)
denote the gradient of the d-variable function f.


### 12.1. Transfinite Lagrange-Hermite Interpolation to a Functionf∈C1,d from B1,d on the Diagonal Δ* of Id

Theorem 12.1.
Letf∈C1,d. Then we have the following(a)
The functionhf*∈B1,d, where
(12.5)hf*(x1,…,xd):=f(-1,…,-1)+∑i=1d∫-1xi∂f∂xi(t,…,t)dt
is the unique transfinite interpolant tof from  B1,d  satisfying  the  following  Lagrange-Hermite transfinite interpolation conditions:
(12.6)(hf*)∣Δ*=f|Δ*,(gradhf*)|Δ*=(gradf)|Δ*.(b)
The following error representation formula holds(12.7)f(x)-hf*(x)=-∑1≤i<j≤d∂2f∂xi∂xj(ξi,j)(xi-xj)22,wherex=(x1,…,xd)∈Id and  ξi,j∈Id.Proof.
Proof of (a). Letx=(x,…,x)∈Δ*. Then (∂hf*/∂xi)(x)=(∂f/∂xi)(x,…,x)=(∂f/∂xi)(x) and, from here, (gradhf*)|Δ*=(gradf)|Δ*.
Proof of (b). Letx=(x1,…,xd)∈Id. Without any restriction (with a permutation of the variables if necessary) we suppose x1≤x2⋯≤xd. Consider the auxiliary function g(x)=f(x)-hf*(x). Then by (12.5), g|Δ*=0 and (gradg)|Δ*=0, and in view of this, for each  i,1≤i≤d(12.8)∂g∂xi(x)=∂g∂xi(x1,…,xd)-∂g∂xi(xi,…,xi)=-∑j=1,j≠id(xi-xj)∫01∂2g∂xi∂xj[x1+s(xi-x1),…,xd+s(xi-xd)]ds.
Denote  xk:=(xk,…,xk,xk+1,…,xd),k=1,…,d, x1=x=(x1,…,xd), xd=(xd,…,xd). By using  the  representation  (12.8)  for  ∂g/∂xi we obtain
(12.9)g(x)=∑k=1d-1[g(xk)-g(xk+1)]=-∑k=1d-1∫xkxk+1ddtg(t,…,t,xk+1,…,xd)dt=-∑k=1d-1∫xkxk+1∑i=1k∂g∂xi(t,…,t,xk+1,…,xd)dt=∑k=1d-1∑i=1k∑j=k+1d∫xkxk+1(t-xj)∫01∂2g∂xi∂xj(xs,t,k)dsdt,
where xs,t,k:=(t,…,t,xk+1+s(t-xk+1),…,xd+s(t-xd)). Changing twice the order of summation in (12.9) and applying the integral mean value theorem we obtain
(12.10)g(x)=∑i=1d-1∑k=id-1∑j=k+1d∫xkxk+1(t-xj)∫01∂2g∂xi∂xj(xs,t,k)dsdt=∑i=1d-1∑j=i+1d∑k=ij-1∫xkxk+1(t-xj)∫01‍∂2g∂xi∂xj(xs,t,k)dsdt=∑i=1d-1∑j=i+1d∑k=ij-1[(xk+1-xj)22-(xk-xj)22]Di,j1,1g(xk,j)=-∑i=1d-1∑j=i+1dDi,j1,1g(ξi,j)(xξ-xj)22,ξi,j∈Id,1≤i<j≤d.
This completes the proof.


### 12.2. Transfinite Cubature Formula, Exact in the Linear SpaceB1,d

Leth∈B1,d. Then the following transfinite cubature formula holds:
(12.11)∫⋯∫Idh(x1,…,xd)dx1⋯dxd=2d-1∫Δ*h(t,…,t)dt.Proof.
Integrate (12.7) on Id for f=h∈B1,d, taking into account that hh*=h if h∈B1,d and the representation (12.5) of hh*.The next theorem gives a canonical set characterization of the best one-sided from aboveL1-approximant from B1,d to a function, belonging to the convex cone
(12.12)𝒞+,d:={f∈C1,d:Di,j1,1f≥0onI2;1≤i<j≤d}.Theorem 12.2.
Letf∈𝒞+,d. Then the unique transfinite Lagrange-Hermite interpolant hf* from B1,d to f satisfying the transfinite interpolation conditions
(12.13)(hf*)∣Δ*=f|Δ*,(gradhf*)|Δ*=(gradf)|Δ*
is the unique best one-sided from above L1-approximant to f from B1,d.Proof.
By (12.7) we conclude that f(x)≤hf*(x),x∈Id. Taking into account the cubature (12.11) we conclude by Theorem 10.1 that the transfinite interpolant hf* to f is the best one-sided from above L1-approximant to f∈𝒞+,d from B1,d.
Uniqueness of the Best One-Sided from above
L
1
-Approximant from
B
1
,
d
to
C
+
,
d.
Supposeh∈B1,d, h≥f on Id is another best one-sided from above L1-approximant to f∈𝒞+,d from B1,d. Then, by using (12.6) and (12.11) we obtain
(12.14)0=∫Id(hf*-f)-∫Id(h-f)=∫Id(hf*-h)=2d-1∫Δ*(hf*-h)=2d-1∫Δ*(f-h)≤0.
Hence, h|Δ*=f|Δ*. However, h≥f on Id and from here, (gradh)|Δ*=(gradf)|Δ*. By Theorem 12.1 we conclude that h=hf*. The proof is completed.Remark 12.3.
According to Theorem12.2, the set Δ* is the canonical point set of best one-sided from above L1-approximation from B1,d to the convex cone 𝒞+,d.
Denote byEd the set
(12.15)Ed:={ɛ=(ɛ1,…,ɛd),ɛi=±1,i=1,…,d}
of all d-dimensional vectors with entries ±1.  For a fixed  ɛ∈Ed consider the ɛ-diagonal of Id: Δɛ:={(ɛ1t,…,ɛdt):t∈[-1,1]}. By using the linear transformation ui=ɛixi,i=1,…,d, the following corollary by Theorems 12.1 and 12.2 holds.Corollary 12.4.
Let, for a fixed vectorɛ=(ɛ1,…,ɛd)∈Ed, the function f∈C1,d satisfy
(12.16)ɛiɛjDi,j1,1f≥0onId,1≤i<j≤d.
Then the unique Lagrange-Hermite interpolant
(12.17)hɛ,f*(x1,…,xd)=f(-ɛ1,…,-ɛd)+∑i=1dɛi∫-1ɛixi∂f∂xi(ɛ1t,…,ɛdt)dt
from B1,d to f, satisfying the transfinite interpolation conditions
(12.18)(hε,f*)|Δε=f|Δε,(gradhε,f*)|Δε=(gradf)|Δε,
is the unique best one-sided from above L1-approximant to f from B1,d.Remark 12.5.
Theorems12.1 and 12.2 are common basis for well-known classical and new inequalities. For example, the inequalities 9, 13, 16, 25, and 61 published in [31] are corollaries from the explicit constructions (12.5) and (12.7) of the best one-sided from above L1-approximants to appropriately chosen functions (see [30] for details). More precisely, the right-hand side expressions of these inequalities are best one-sided from above L1-approximants to the left-hand side ones in the sense of Theorems 12.1 and 12.2. We give some examples.
For a set ofd numbers {x1,…,xd} denote x*:=min{x1,…,xd}, x*:=max{x1,…,xd}.Example 12.6.
Considerf(x1,…,xd)=x1p1⋯xdpd, where xi>0,pi>0,i=1,…,d and ∑i=1d‍pi=1. We compute
(12.19)∂f∂xi(x)=pixipi-1∏j=1,j≠idxjpj;∂2f∂xi∂xj(x)=pixipi-1pjxjpj-1∏k=1,k≠i,jdxkpk>0
for xi>0,i=1,…,d. Then
(12.20)hf*(x1,…,xd)=f(x*,…,x*)+∑i=1d∫x*xi∂f∂xi(t,…,t)dt=∑i=1d∫0xipitpi-1∏j=1,j≠idtpjdt=∑i=1d∫0xipidt=∑i=1dpixi
is the unique best one-sided from above L1-approximant to f from B1,d on [x*,x*]d. Hence,
(12.21)x1p1⋯xdpd≤p1x1+⋯+pdxd
with the case of equality only for x1=⋯=xd. The inequality is easily extended to xi≥0,i=1,…,d. This is the well-known inequality between the geometric mean and the arithmetic mean of a set of nonnegative numbers.Example 12.7.
Letf(t) be a univariate function satisfying f′(t)+tf′′(t)>0 for t>0. Then, for the d-variable function f(x1⋯xd),xi>0,i=1,…,d  we compute
(12.22)∂f∂xi(x)=[∏j=1,j≠idxj]f′(x1⋯xd)∂2f∂xi∂xj(x)=[∏k=1,k≠i,jdxk]f′(x1⋯xd)+[∏k=1,k≠idxk][∏s=1,s≠jdxs]f′′(x1⋯xd)=[∏k=1,k≠i,jdxk](f′(x1⋯xd)+(x1⋯xd)f′′(x1⋯xd))>0
for xi>0,i=1,…,d. In view of Theorems 12.1 and 12.2 the unique transfinite interpolant to f from  B1,d on  [x*,x*]2(12.23)hf*(x1,…,xd)=f(x*)+∑i=1d∫x*xi∂f∂xi(t,…,t)dt=f(x*)+∑i=1d∫x*xitd-1f′(td)dt=1d∑i=1df(xid)
is the unique best one-sided from above L1-approximant to f from B1,d on [x*,x*]d. Hence,
(12.24)f(x1⋯xd)≤1d∑i=1df(xid)
with the case of equality only for x1=⋯=xd.Example 12.8.
Letf(t)=(β+t)α,β≥0,α>0,t>0.  We compute
(12.25)f′(t)+tf′′(t)=(β+t)α-2α(β+αt)>0.
Then as a corollary by Example 12.7, we obtain the inequality
(12.26)(β+x1⋯xd)a≤1d∑i=1d(β+xid)α,xi>0,i=1,…,d
with the case of equality only for x1=⋯=xd. Obviously, the inequality holds for xi≥0,i=1,…,d.Example 12.9.
Letf(t)=sin(t),t≥0. We compute f′(t)+tf′′(t)=cos(t)-tsin(t)>0 for t∈(0,t*), where t*≈0.8603336 is the unique solution of the nonlinear equation cot(t)=t,t∈[0,π/2]. Then, following Example 12.7, we obtain the inequality
(12.27)sin(x1⋯xd)≤1d∑i=1dsin(xid),0<xi<t*1/d,1≤i≤d
with the case of equality only for x1=⋯=xd. Obviously the inequality holds for 0≤xi≤t*1/d,1≤i≤d.Example 12.10.
Let the univariate functionf(t) satisfy 2f′(t)+tf′′(t)>0 for t>0. Consider the d-variable function
(12.28)f~(x1,…,xd):=∑1≤i<j≤df(1xi+xj),xi>0,1≤i≤d.
Computing partial derivatives of f~ we obtain
(12.29)∂f~∂xi=-∑j=1,j≠id1(xi+xj)2f′(1xi+xj)∂2f~∂xi∂xj=2(xi+xj)3f′(1xi+xj)+1(xi+xj)4f′′(1xi+xj)=1(xi+xj)3[2f′(1xi+xj)+1(xi+xj)f′′(1xi+xj)]>0,
where 1≤i<j≤d. In view of Theorems 12.1 and 12.2, the transfinite interpolant to f~ from B1,d on [x*,x*]d(12.30)hf~*(x1,…,xd)=f~(x*,…,x*)+∑i=1d∂f~∂xi(t,…,t)dt=d(d-1)2f(12x*)-(d-1)∑i=1d∫x*xi14t2f′(12t)dt=d-12∑i=1df(12xi)
is the unique best one-sided from above L1-approximant to f~ from B1,d. Hence, the following inequality holds:
(12.31)∑1≤i<j≤df(1xi+xj)≤d-12∑i=1df(12xi)
with the case of equality only for x1=⋯=xd. In the particular cases f(t)=t and f(t)=ln(t)  we obtain the inequalities
(12.32)∑1≤i<j≤d1xi+xj≤d-14∑i=1d1xi;∑1≤i<j≤dln1xi+xj≤d-12∑i=1dln12xi
with the case of equalities only for  x1=⋯=xd.


## 12.1. Transfinite Lagrange-Hermite Interpolation to a Functionf∈C1,d from B1,d on the Diagonal Δ* of Id

Theorem 12.1.
Letf∈C1,d. Then we have the following(a)
The functionhf*∈B1,d, where
(12.5)hf*(x1,…,xd):=f(-1,…,-1)+∑i=1d∫-1xi∂f∂xi(t,…,t)dt
is the unique transfinite interpolant tof from  B1,d  satisfying  the  following  Lagrange-Hermite transfinite interpolation conditions:
(12.6)(hf*)∣Δ*=f|Δ*,(gradhf*)|Δ*=(gradf)|Δ*.(b)
The following error representation formula holds(12.7)f(x)-hf*(x)=-∑1≤i<j≤d∂2f∂xi∂xj(ξi,j)(xi-xj)22,wherex=(x1,…,xd)∈Id and  ξi,j∈Id.Proof.
Proof of (a). Letx=(x,…,x)∈Δ*. Then (∂hf*/∂xi)(x)=(∂f/∂xi)(x,…,x)=(∂f/∂xi)(x) and, from here, (gradhf*)|Δ*=(gradf)|Δ*.
Proof of (b). Letx=(x1,…,xd)∈Id. Without any restriction (with a permutation of the variables if necessary) we suppose x1≤x2⋯≤xd. Consider the auxiliary function g(x)=f(x)-hf*(x). Then by (12.5), g|Δ*=0 and (gradg)|Δ*=0, and in view of this, for each  i,1≤i≤d(12.8)∂g∂xi(x)=∂g∂xi(x1,…,xd)-∂g∂xi(xi,…,xi)=-∑j=1,j≠id(xi-xj)∫01∂2g∂xi∂xj[x1+s(xi-x1),…,xd+s(xi-xd)]ds.
Denote  xk:=(xk,…,xk,xk+1,…,xd),k=1,…,d, x1=x=(x1,…,xd), xd=(xd,…,xd). By using  the  representation  (12.8)  for  ∂g/∂xi we obtain
(12.9)g(x)=∑k=1d-1[g(xk)-g(xk+1)]=-∑k=1d-1∫xkxk+1ddtg(t,…,t,xk+1,…,xd)dt=-∑k=1d-1∫xkxk+1∑i=1k∂g∂xi(t,…,t,xk+1,…,xd)dt=∑k=1d-1∑i=1k∑j=k+1d∫xkxk+1(t-xj)∫01∂2g∂xi∂xj(xs,t,k)dsdt,
where xs,t,k:=(t,…,t,xk+1+s(t-xk+1),…,xd+s(t-xd)). Changing twice the order of summation in (12.9) and applying the integral mean value theorem we obtain
(12.10)g(x)=∑i=1d-1∑k=id-1∑j=k+1d∫xkxk+1(t-xj)∫01∂2g∂xi∂xj(xs,t,k)dsdt=∑i=1d-1∑j=i+1d∑k=ij-1∫xkxk+1(t-xj)∫01‍∂2g∂xi∂xj(xs,t,k)dsdt=∑i=1d-1∑j=i+1d∑k=ij-1[(xk+1-xj)22-(xk-xj)22]Di,j1,1g(xk,j)=-∑i=1d-1∑j=i+1dDi,j1,1g(ξi,j)(xξ-xj)22,ξi,j∈Id,1≤i<j≤d.
This completes the proof.


## 12.2. Transfinite Cubature Formula, Exact in the Linear SpaceB1,d

Leth∈B1,d. Then the following transfinite cubature formula holds:
(12.11)∫⋯∫Idh(x1,…,xd)dx1⋯dxd=2d-1∫Δ*h(t,…,t)dt.Proof.
Integrate (12.7) on Id for f=h∈B1,d, taking into account that hh*=h if h∈B1,d and the representation (12.5) of hh*.The next theorem gives a canonical set characterization of the best one-sided from aboveL1-approximant from B1,d to a function, belonging to the convex cone
(12.12)𝒞+,d:={f∈C1,d:Di,j1,1f≥0onI2;1≤i<j≤d}.Theorem 12.2.
Letf∈𝒞+,d. Then the unique transfinite Lagrange-Hermite interpolant hf* from B1,d to f satisfying the transfinite interpolation conditions
(12.13)(hf*)∣Δ*=f|Δ*,(gradhf*)|Δ*=(gradf)|Δ*
is the unique best one-sided from above L1-approximant to f from B1,d.Proof.
By (12.7) we conclude that f(x)≤hf*(x),x∈Id. Taking into account the cubature (12.11) we conclude by Theorem 10.1 that the transfinite interpolant hf* to f is the best one-sided from above L1-approximant to f∈𝒞+,d from B1,d.
Uniqueness of the Best One-Sided from above
L
1
-Approximant from
B
1
,
d
to
C
+
,
d.
Supposeh∈B1,d, h≥f on Id is another best one-sided from above L1-approximant to f∈𝒞+,d from B1,d. Then, by using (12.6) and (12.11) we obtain
(12.14)0=∫Id(hf*-f)-∫Id(h-f)=∫Id(hf*-h)=2d-1∫Δ*(hf*-h)=2d-1∫Δ*(f-h)≤0.
Hence, h|Δ*=f|Δ*. However, h≥f on Id and from here, (gradh)|Δ*=(gradf)|Δ*. By Theorem 12.1 we conclude that h=hf*. The proof is completed.Remark 12.3.
According to Theorem12.2, the set Δ* is the canonical point set of best one-sided from above L1-approximation from B1,d to the convex cone 𝒞+,d.
Denote byEd the set
(12.15)Ed:={ɛ=(ɛ1,…,ɛd),ɛi=±1,i=1,…,d}
of all d-dimensional vectors with entries ±1.  For a fixed  ɛ∈Ed consider the ɛ-diagonal of Id: Δɛ:={(ɛ1t,…,ɛdt):t∈[-1,1]}. By using the linear transformation ui=ɛixi,i=1,…,d, the following corollary by Theorems 12.1 and 12.2 holds.Corollary 12.4.
Let, for a fixed vectorɛ=(ɛ1,…,ɛd)∈Ed, the function f∈C1,d satisfy
(12.16)ɛiɛjDi,j1,1f≥0onId,1≤i<j≤d.
Then the unique Lagrange-Hermite interpolant
(12.17)hɛ,f*(x1,…,xd)=f(-ɛ1,…,-ɛd)+∑i=1dɛi∫-1ɛixi∂f∂xi(ɛ1t,…,ɛdt)dt
from B1,d to f, satisfying the transfinite interpolation conditions
(12.18)(hε,f*)|Δε=f|Δε,(gradhε,f*)|Δε=(gradf)|Δε,
is the unique best one-sided from above L1-approximant to f from B1,d.Remark 12.5.
Theorems12.1 and 12.2 are common basis for well-known classical and new inequalities. For example, the inequalities 9, 13, 16, 25, and 61 published in [31] are corollaries from the explicit constructions (12.5) and (12.7) of the best one-sided from above L1-approximants to appropriately chosen functions (see [30] for details). More precisely, the right-hand side expressions of these inequalities are best one-sided from above L1-approximants to the left-hand side ones in the sense of Theorems 12.1 and 12.2. We give some examples.
For a set ofd numbers {x1,…,xd} denote x*:=min{x1,…,xd}, x*:=max{x1,…,xd}.Example 12.6.
Considerf(x1,…,xd)=x1p1⋯xdpd, where xi>0,pi>0,i=1,…,d and ∑i=1d‍pi=1. We compute
(12.19)∂f∂xi(x)=pixipi-1∏j=1,j≠idxjpj;∂2f∂xi∂xj(x)=pixipi-1pjxjpj-1∏k=1,k≠i,jdxkpk>0
for xi>0,i=1,…,d. Then
(12.20)hf*(x1,…,xd)=f(x*,…,x*)+∑i=1d∫x*xi∂f∂xi(t,…,t)dt=∑i=1d∫0xipitpi-1∏j=1,j≠idtpjdt=∑i=1d∫0xipidt=∑i=1dpixi
is the unique best one-sided from above L1-approximant to f from B1,d on [x*,x*]d. Hence,
(12.21)x1p1⋯xdpd≤p1x1+⋯+pdxd
with the case of equality only for x1=⋯=xd. The inequality is easily extended to xi≥0,i=1,…,d. This is the well-known inequality between the geometric mean and the arithmetic mean of a set of nonnegative numbers.Example 12.7.
Letf(t) be a univariate function satisfying f′(t)+tf′′(t)>0 for t>0. Then, for the d-variable function f(x1⋯xd),xi>0,i=1,…,d  we compute
(12.22)∂f∂xi(x)=[∏j=1,j≠idxj]f′(x1⋯xd)∂2f∂xi∂xj(x)=[∏k=1,k≠i,jdxk]f′(x1⋯xd)+[∏k=1,k≠idxk][∏s=1,s≠jdxs]f′′(x1⋯xd)=[∏k=1,k≠i,jdxk](f′(x1⋯xd)+(x1⋯xd)f′′(x1⋯xd))>0
for xi>0,i=1,…,d. In view of Theorems 12.1 and 12.2 the unique transfinite interpolant to f from  B1,d on  [x*,x*]2(12.23)hf*(x1,…,xd)=f(x*)+∑i=1d∫x*xi∂f∂xi(t,…,t)dt=f(x*)+∑i=1d∫x*xitd-1f′(td)dt=1d∑i=1df(xid)
is the unique best one-sided from above L1-approximant to f from B1,d on [x*,x*]d. Hence,
(12.24)f(x1⋯xd)≤1d∑i=1df(xid)
with the case of equality only for x1=⋯=xd.Example 12.8.
Letf(t)=(β+t)α,β≥0,α>0,t>0.  We compute
(12.25)f′(t)+tf′′(t)=(β+t)α-2α(β+αt)>0.
Then as a corollary by Example 12.7, we obtain the inequality
(12.26)(β+x1⋯xd)a≤1d∑i=1d(β+xid)α,xi>0,i=1,…,d
with the case of equality only for x1=⋯=xd. Obviously, the inequality holds for xi≥0,i=1,…,d.Example 12.9.
Letf(t)=sin(t),t≥0. We compute f′(t)+tf′′(t)=cos(t)-tsin(t)>0 for t∈(0,t*), where t*≈0.8603336 is the unique solution of the nonlinear equation cot(t)=t,t∈[0,π/2]. Then, following Example 12.7, we obtain the inequality
(12.27)sin(x1⋯xd)≤1d∑i=1dsin(xid),0<xi<t*1/d,1≤i≤d
with the case of equality only for x1=⋯=xd. Obviously the inequality holds for 0≤xi≤t*1/d,1≤i≤d.Example 12.10.
Let the univariate functionf(t) satisfy 2f′(t)+tf′′(t)>0 for t>0. Consider the d-variable function
(12.28)f~(x1,…,xd):=∑1≤i<j≤df(1xi+xj),xi>0,1≤i≤d.
Computing partial derivatives of f~ we obtain
(12.29)∂f~∂xi=-∑j=1,j≠id1(xi+xj)2f′(1xi+xj)∂2f~∂xi∂xj=2(xi+xj)3f′(1xi+xj)+1(xi+xj)4f′′(1xi+xj)=1(xi+xj)3[2f′(1xi+xj)+1(xi+xj)f′′(1xi+xj)]>0,
where 1≤i<j≤d. In view of Theorems 12.1 and 12.2, the transfinite interpolant to f~ from B1,d on [x*,x*]d(12.30)hf~*(x1,…,xd)=f~(x*,…,x*)+∑i=1d∂f~∂xi(t,…,t)dt=d(d-1)2f(12x*)-(d-1)∑i=1d∫x*xi14t2f′(12t)dt=d-12∑i=1df(12xi)
is the unique best one-sided from above L1-approximant to f~ from B1,d. Hence, the following inequality holds:
(12.31)∑1≤i<j≤df(1xi+xj)≤d-12∑i=1df(12xi)
with the case of equality only for x1=⋯=xd. In the particular cases f(t)=t and f(t)=ln(t)  we obtain the inequalities
(12.32)∑1≤i<j≤d1xi+xj≤d-14∑i=1d1xi;∑1≤i<j≤dln1xi+xj≤d-12∑i=1dln12xi
with the case of equalities only for  x1=⋯=xd.



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*Source: 435945-2012-10-04.xml*