# Best Approximation of Data Distributed in a Band

Authors: H. Cuenya; F. Levis; C. Rodriguez
Journal: ISRN Mathematical Analysis
 (2011)

---

## Abstract

We study the problem to approximate a data set which are affected in a such way that they present us as a band in the plane. We introduce
a deviation measure, and we research the asymptotic behavior of the best approximants when the band shrink in some sense.


---

## Body

## 1. Introduction

In some situations, we find us with the problem to approximate a given function of physical origin which is contaminated by different causes. For example, it occurs when we receive a signal, and we observe at the screen of an electronic oscilloscope a band produced by noise or other factors. Here, a criterion of selecting is necessary in order to approximate to that band. More precisely, we must choose a measure of deviation from one band to a given approximant class. A way could be to approximate a segment value multivalued function using the Hausdorff metric in the plane (see [1]); another could be to consider the best simultaneous approximation to the set of every functions whose graphics live in the band determined by them (see [2–4]). In this paper, we give an alternative deviation measure, and we establish a relation with the best simultaneous approximation.Let1≤p<∞, and F:[a,b]→2ℝ, be a multivalued function with F(x) a Lebesgue measurable set for all x∈[a,b]. Given an approximant class 𝒮, we consider the following function as measure of deviation of F to Q∈𝒮:(1.1)Φ[a,b](F,Q)=(∫ab∫F(x)|y-Q(x)|pdydx)1/p.LetD:={(x,y):x∈[a,b],y∈F(x)}. It is easy to see that (1.1) is a special case to approximate a function I from a given class 𝒞 with a norm |∥·∥| over the space of functions defined on D. In fact, if 𝒞={HQ:Q∈𝒮}, where HQ(x,y)=Q(x), then Φ[a,b](F,Q)=|∥I-HQ∥|, where I(x,y)=y and(1.2)|‖G‖|=(∬D|G(x,y)|pdydx)1/p.As usual, if1≤q≤∞, A⊂[a,b] is a Lebesgue measurable set and w is a nonnegative integrable function on A, then  Lwq(A) denotes the space of Lebesgue measurable functions f satisfying (1.3)‖f‖q,w,A:=(∫A|f(x)|qw(x)dx)1/q<∞,
with the usual understanding if q=∞. If w=1, we write Lq(A) and ∥f∥q,A for Lwq(A) and ∥f∥q,w,A, respectively.Given two functionsf,g:[a,b]→ℝ, f≤g, we only consider in this paper the multivalued function F defined by F(x)=[f(x),g(x)] for each x∈[a,b].Our main goal is to study the asymptotic behavior of thoseQ which minimize (1.1)
when the band shrinks in some sense, and the approximant class𝒮 is a finite dimensional linear subspace. In this case, 𝒞 is a finite dimensional linear subspace, and the existence of such a Q is well known (see [5]).We consider that the band shrinks to a curve in two situations:(i)
The functionsf and g are replaced by a family of functions fϵ, gϵ, where fϵ, gϵ converge to a function h, as ϵ tends to 0. That is, the band shrinks vertically.(ii)
The interval[a,b] is substituted by [x0-ϵ,x0+ϵ], where ϵ tends to 0; that is, the band shrinks horizontally.If there exists the limit ofQ minimizing (1.1) when the band shrinks to a curve, as such, it provides useful qualitative and approximation analytic information concerning the approximants on small bands, which is difficult to obtain from a strictly numerical treatment. The existence of the limit of Q is close to the best local approximation problem (see [6–8]).In Section2, we prove that if the band shrinks vertically to a given function, then the set of closure points of Q is contained in the set of best approximants to that function, with a suitable seminorm. Moreover, we see that the limit of Q exists when p>1.In Section3, we prove that if the band shrinks horizontally, the limit of Q is the mean of the Taylor polynomials of f and g at x0. We also show that this approximation problem is related with the subject of best simultaneous local approximation which was studied in [8].We assume conditions about the functionsf and g in order to be (1.1) finite for all Q∈𝒮. Henceforward, f,g∈Lq([a,b]),𝒮⊂Lq([a,b]), and g-f∈Lq/(q-p)([a,b]),p≤q. In this case, using Hölder's inequality, we have(1.4)Φ[a,b]p(F,Q)≤∫abmax⁡{|f(x)-Q(x)|,|g(x)-Q(x)|}p(g(x)-f(x))dx≤‖max⁡{|f-Q|,|g-Q|}‖q,[a,b]p‖g-f‖q/(q-p),[a,b]<∞,
for all Q∈𝒮. If q≥p+1, the condition g-f∈Lq/(q-p)([a,b]) is automatically satisfied.


## 2. The Band Shrinks Vertically

Letg be a measurable nonnegative function in Lq([a,b]), p≤q, and {fϵ}, {gϵ}, ϵ>0, two net of measurable functions such that |fϵ|,|gϵ|≤g, a.e. on [a,b]. We write Fϵ(x)=[fϵ(x),gϵ(x)], x∈[a,b]. Given 𝒮⊂Lq([a,b]) a finite dimensional lineal subspace, let Qϵ∈𝒮 which minimizes Φ[a,b](Fϵ,Q), Q∈𝒮.Theorem 2.1.
Assume that there are two functionsW(ϵ):(0,∞)→(0,∞) and h∈Lq/(q-p)([a,b]) such that almost everywhere on [a,b](2.1)gϵ-fϵW(ϵ)≤h,(2.2)lim⁡ϵ→0gϵ-fϵW(ϵ)=w,
with |{w>0}|>0. If fϵ and gϵ converge to f almost everywhere on [a,b], then the set of closure points of Qϵ is a nonempty set, and it is contained in the set of best approximants to f from 𝒮 with the seminorm ∥·∥p,w,[a,b]. In particular, if p>1, the net Qϵ converges to the unique best approximant to f, as ϵ→0.Proof.
Forϵ>0, we denote wϵ*=wϵ/W(ϵ), where wϵ=gϵ-fϵ. Let
(2.3)JQ,ϵ(x):=∫fϵ(x)gϵ(x)|y-Q(x)|pdy,x∈[a,b],Q∈S.
Let Q∈𝒮. By integral mean value theorem, for each x∈[a,b], there exists
(2.4)αϵ(x),ξϵ(x)∈[fϵ(x),gϵ(x)],
such that JQϵ,ϵ=|αϵ-Qϵ|pwϵ and JQ,ϵ=|ξϵ-Q|pwϵ. Now, the Fubbini Theorem implies that JQϵ,ϵ and JQ,ϵ are measurable functions on [a,b], and hence, αϵ and ξϵ are measurable functions on [a,b]. Consequently, αϵ, ξϵ∈Lq([a,b]), because |αϵ|,|ξϵ|≤g, a.e. on [a,b].
On the other hand,(2.5)‖f-Qϵ‖p,wϵ,[a,b]-‖αϵ-f‖p,wϵ,[a,b]≤‖αϵ-Qϵ‖p,wϵ,[a,b]=Φ[a,b](Fϵ,Qϵ)≤Φ[a,b](Fϵ,Q)=‖ξϵ-Q‖p,wϵ,[a,b].
In consequence, we get
(2.6)‖f-Qϵ‖p,wϵ*,[a,b]≤‖αϵ-f‖p,wϵ*,[a,b]+‖ξϵ-Q‖p,wϵ*,[a,b].
As |f|≤g a.e. on [a,b], from (2.1) and the Hölder inequality, we have
(2.7)|αϵ-f|pwϵ*≤(2g)ph∈L1([a,b]),|ξϵ-Q|pwϵ*≤(g+|Q|)ph∈L1([a,b]).
From (2.2) and the Lebesgue Dominated Convergence Theorem, it follows that
(2.8)lim⁡ϵ→0(‖αϵ-f‖p,wϵ*,[a,b]+‖ξϵ-Q‖p,wϵ*,[a,b])=‖f-Q‖p,w,[a,b].
By hypothesis, there is k>0 satisfying |{w>k}|>0. By the Egoroff Theorem (see [9]), there exists a set A⊂{w>k}, |A|>0, where we have that wϵ* uniformly converges to w on A. So, we can choose a positive constant ϵ0 such that for all x∈A and all 0<ϵ<ϵ0, wϵ*(x)≥k/2. Hence,
(2.9)‖f-Qϵ‖p,A≤(k2)-1/p‖f-Qϵ‖p,wϵ*,[a,b],0<ϵ<ϵ0.
According to (2.6)–(2.9), {Qϵ} is a uniformly bounded net. Then, {Qϵ} have a subsequence that we again denote by {Qϵ} converging to T∈𝒮. Again, the Lebesgue dominated convergence theorem implies limϵ→0∥f-Qϵ∥p,wϵ*,[a,b]=∥f-T∥p,w,[a,b]. Finally, from (2.6) and (2.8), we conclude that
(2.10)‖f-T‖p,w,[a,b]≤‖f-Q‖p,w,[a,b].
As Q∈𝒮 is arbitrary, the theorem immediately follows.Suppose thatp>1. If we have two functions Wi:(0,∞)→(0,∞) and wi, i=1,2, fulfilling the hypothesis of Theorem 2.1, we conclude that Qϵ converges to the best approximant to f from 𝒮 when we consider ∥·∥p,w1,[a,b] and ∥·∥p,w2,[a,b], respectively. The next lemma shows that it is not surprising, because these norms differing by a constant.Lemma 2.2.
Let(Ω,Σ,μ) be a finite measure space. Let wϵ:Ω→ℝ,ϵ>0 be a net of nonnegative measurable functions. Assume that there are two functions Wi:(0,∞)→(0,∞), i=1,2 such that almost everywhere on Ω(2.11)lim⁡ϵ→0wϵ(x)Wi(ϵ)=wi(x)
is finite. If Ai:={x∈Ω:wi(x)>0} and μ(Ai)>0, i=1,2, then there exists k>0 satisfying w1=kw2, a.e. on Ω.Proof.
We denote byIi, i=1,2, the subsets of Ω, where wi is finite, and we write K=I1∩I2. Clearly, μ(K)=μ(Ω). Let Ji=Ai∩K.
Ifμ(J1∩J2)=0, then μ(J1-J2)=μ(A1)>0 and μ(J2-J1)=μ(A2)>0. We take x1∈J1-J2 and x2∈J2-J1. The equality
(2.12)wϵ(x)W2(ϵ)=wϵ(x)W1(ϵ)W1(ϵ)W2(ϵ),x∈Ω
implies that W1(ϵ)/W2(ϵ) tends to zero if x=x1, and it tends to infinite if x=x2, a contradiction. So, we get μ(J1∩J2)>0.
For allx∈J1∩J2, we have
(2.13)0<k:=lim⁡ϵ→0W1(ϵ)W2(ϵ)=w2(x)w1(x).
Next, we prove that μ(J1-J2)=μ(J2-J1)=0. If μ(J1-J2)>0 the above argument implies that k=0, and if μ(J2-J1)>0, we obtain k=∞. In either case, it contradicts (2.13). Therefore, μ(J1∩J2)=μ(J1∪J2)=μ(A1)=μ(A2) and
(2.14)w2(x)=kw1(x),a.e. onA1∪A2.
It immediately follows that (2.14) holds a.e. on Ω.We also observe in the next example that (2.2) in Theorem 2.1 is essential.Example 2.3.
Leta=0, b=1, and p=2. We consider 𝒮 the space of constant functions,
(2.15)fm(x)=x,gm={x+1m,ifmiseven,x+xm,ifmisodd.
Here, m(gm-fm), is equal to 1 if m is even, and it is equal x in otherwise. Applying Theorem 2.1, we get Q2m→1/2 and Q2m-1→2/3, as m→∞.


## 3. The Band Shrinks Horizontally

Letx0∈[a,b] and Iϵ:=[x0-ϵ,x0+ϵ]⊂[a,b], ϵ>0. In this section, we assume that f and g have continuous derivatives up to order n at x0. From now on, 𝒮=Πn is the space of algebraic polynomials of degree at most n, and Qϵ denotes an element of Πn that minimizes ΦIϵ(F,Q), Q∈Πn.We recall (see [10, 11]) the Newton divided difference formula for the interpolation polynomial of a function h of degree n at x1≤x2≤⋯≤xn+1,(3.1)P(x)=h(x1)+(x-x1)h[x1,x2]+⋯+(x-x1)⋯(x-xn)h[x1,…,xn+1].
Here, h[x1,…,xm+1] denotes the m-th order Newton divided difference. If h has continuous derivatives up to order m, on an interval [a,b] containing to x1,…,xm+1, then the m-th divided difference can be expressed as(3.2)h[x1,…,xm+1]=h(m)(ξ)m!,
for some ξ in the interval [x1,xm+1]. It is well known that the m-th divided difference is a continuous function as function of their arguments x1,…,xm+1.For simplicity of notation, we writeT(f) and T(g) the Taylor polynomials of f and g at x0 of order n, respectively.Lemma 3.1.
Letq≥0, and let fi:[a,b]→ℝ, 1≤i≤3 be continuous functions with f1≤f2. If
(3.3)h(x)=∫f1(x)f2(x)|y-f3(x)|qsgn⁡(y-f3(x))dy,a≤x≤b,
then h is a continuous function.Proof.
Ifq>0, the continuity of h follows from the continuity of fi, 1≤i≤3 and the uniform continuity of tqsgn(t) on compact sets. If q=0, we have
(3.4)h(x)={f1(x)-f2(x),iff3(x)≥f2(x),f2(x)-f1(x),iff3(x)≤f1(x),f1(x)+f2(x)-2f3(x),iff1(x)≤f3(x)≤f2(x).
Thus, the continuity of h is immediate.Theorem 3.2.
Ifp≥1, then any net Qϵ converges to (T(f)+T(g))/2, as ϵ→0.Proof.
It is easy to see thatQϵ is characterized by
(3.5)∫Iϵh(x)Q(x)dx=0,Q∈Πn,
where
(3.6)h(x)=∫f(x)g(x)|y-Qϵ(x)|p-1sgn⁡(y-Qϵ(x))dy.
By Lemma 3.1, h is continuous. Therefore, h interpolates to zero in at least n+1 different points of the interval Iϵ, say x1<x2<⋯<xn+1. In fact, if h has m different zeros, m≤n, we can find an element Q∈Πn such that h(x)Q(x)>0 for all x≠xi, 1≤i≤m. It contradicts (3.5). So,
(3.7)∫f(xi)g(xi)|y-Qϵ(xi)|p-1sgn⁡(y-Qϵ(xi))dy=0,1≤i≤n+1.
It follows that Qϵ(xi) is a best constant approximant to the identity function with norm ∥·∥p,[f(xi),g(xi)], 1≤i≤n+1. A straightforward computation shows that Qϵ(xi)=(f(xi)+g(xi))/2, 1≤i≤n+1; that is, Qϵ interpolates to the function  (f+g)/2 at xi, 1≤i≤n+1. From (3.1) and (3.2), it follows that
(3.8)Qϵ(x)=h(x1)+(x-x1)h(1)(ξ1)+⋯+(x-x1)⋯(x-xn)h(n)(ξn)n!,
where h=(f+g)/2, ξi∈Iϵ, 1≤i≤n. Taking limit for ϵ→0 in (3.8) and using the continuity of the derivatives of the functions f and g, we get the theorem.Corollary 3.3.
Letp≥1. Suppose that f and g have continuous derivatives up to order n+1 at x0 and f(x0)<g(x0). Then, for sufficiently small ϵ, Qϵ is the best lp+1-simultaneous approximant in Lp+1(Iϵ); that is, Qϵ minimizes
(3.9)(‖f-Q‖p+1,Iϵp+1+‖g-Q‖p+1,Iϵp+1)1/(p+1),Q∈Πn.Proof.
By hypothesis, there isϵ0>0 such that f<g on Iϵ0. Let Pϵ be the best lp+1-simultaneous approximant to f and g in Lp+1(Iϵ). From Theorem 3.2, and [8, Theorem  3.4], there exists 0<ϵ1<ϵ0 such that
(3.10)Qϵ(x),Pϵ(x)∈[f(x),g(x)],x∈Iϵ,ϵ<ϵ1.
On the other hand, we have
(3.11)(p+1)ΦIϵp(F,Q)=∫Iϵ|f(x)-Q(x)|p+1sgn⁡(Q(x)-f(x))dx+∫Iϵ|g(x)-Q(x)|p+1sgn⁡(g(x)-Q(x))dx,Q∈Πn,
From (3.10) and (3.11), we get
(3.12)‖f-Qϵ‖p+1,Iϵp+1+‖g-Qϵ‖p+1,Iϵp+1=‖f-Pϵ‖p+1,Iϵp+1+‖g-Pϵ‖p+1,Iϵp+1,
for all ϵ<ϵ1. So, Qϵ=Pϵ for all ϵ<ϵ1.Remark 3.4.
IfΦ[a,b](F,Q)=supx∈[a,b]supy∈F(x)|y-Q(x)|, then
(3.13)Φ[a,b](F,Q)=sup⁡x∈[a,b]max⁡{|f(x)-Q(x)|,|g(x)-Q(x)|}
implies that Qϵ is the best l∞-simultaneous approximant in L∞([a,b]), for all ϵ>0.



---

*Source: 147026-2011-09-13.xml*