pinecone/core-2020-05-10-deduplication huggingface dataset:
Approximation of the Inverse Frame Operator and Applications to Gabor Frames

AbstractA frame allows every element in a Hilbert space H to be written as a linear combination of the frame elements, with coefficients called frame coefficients. Calculation of the frame coefficients requires inversion of an operator S on H. We show how the inverse of S can be approximated as close as we like using finite-dimensional linear algebra. In contrast with previous methods, our approximation can be used for any frame. Various consequences for approximation of the frame coefficients or approximation of the solution to a moment problem are discussed. We also apply the results to Gabor frames and frames consisting of translates of a single function
Universal Functions on Complex General Linear Groups

AbstractIn 1929, Birkhoff proved the existence of an entire function F on C with the property that for any entire function f there exists a sequence {ak} of complex numbers such that {F(ζ+ak)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoff's theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group C*. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n, C) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n, C) be the general linear group of degree n over C. We denote by A(G) the set of all holomorphic functions on G. Similarly, we define A(C). Let K be the A(G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n, C) with f(0)∉(f(K)), where (f(K)) is the A(C)-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any K∈B(G), for any function f holomorphic in some neighbourhood of K, and for any ε>0, there exists C∈G with maxZ∈K|F(CZ)−f(Z)|<ε
The moment problem for a Sobolev inner product

17 pages, no figures.-- MSC2000 code: 44A60.MR#: MR1715024 (2000m:44013)Zbl#: Zbl 0980.44008The concepts of definite and determinate Sobolev moment problem are introduced. The study of these questions is reduced to the definiteness or determinacy, respectively, of a system of classical moment problems by means of a canonical decomposition of the moment matrix associated with a Sobolev inner product in terms of Hankel matrices.Research by second author (G.L.L.) was partially supported by Dirección General de Enseñanza Superior under Grant PB 96-0120-C03-01 and by INTAS under Grant 93-0219 EXT. Research by the third author (H.P.) was carried out while following a Doctoral Program at Universidad Carlos III de Madrid under a grant from Agencia Española de Cooperación Internacional.Publicad
On Certain Mean Values of Polynomials on the Unit Interval

AbstractFor any continuous function f:[−1, 1]↦C and any p∈(0, ∞), let ‖f‖p≔(2−1∫1−1|f(x)|pdx)1/p; in addition, let ‖f‖∞≔max−1⩽x⩽1|f(x)|. It is known that if f is a polynomial of degree n, then for all p>0,‖f‖∞⩽Cpn2/p‖f‖p,where Cp is a constant depending on p but not on n. In this result of Nikolskiı (1951), which was independently obtained by Szegö and Zygmund (1954), the order of magnitude of the bound is the best possible. We obtain a sharp version of this inequality for polynomials not vanishing in the open unit disk. As an application we prove the following result. If f is a real polynomial of degree n such that f(−1)=f(1)=0 and f(z)≠0 in the open unit disk, then for p>0 the quantity ‖f′‖∞/‖f‖p is maximized by polynomials of the form c(1+x)n−1(1−x), c(1+x)(1−x)n−1, where c∈R\{0}. This extends an inequality of Erdo&#x030B;s (1940)
Remarks Concerning Completeness of Translates in Function Spaces

AbstractThis note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k∈Z+ are complete in the spaces Cl0(R), Lp(R), Wlp(R), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues
Weak Type Inequalities for Best Simultaneous Approximation in Banach Spaces

AbstractFor arbitrary Banach spaces Butzer and Scherer in 1968 showed that the approximation order of best approximation can characterized by the order of certain K-functionals. This general theorem has many applications such as the characterization of the best approximation of algebraic polynomials by moduli of smoothness involving the Legendre, Chebyshev, or more general the Jacobi transform. In this paper we introduce a family of seminorms on the underlying approximation space which leads to a generalization of the Butzer–Scherer theorems. Now the characterization of the weighted best algebraic approximation in terms of the so-called main part modulus of Ditzian and Totik is included in our frame as another particular application. The goal of the paper is to show that for the characterization of the orders of best approximation, simultaneous approximation (in different spaces), reduction theorems, and K-functionals one has (essentially) only to verify three types of inequalities, namely inequalities of Jackson-, Bernstein-type and an equivalence condition which guarantees the equivalence of the seminorm and the underlying norm on certain subspaces. All the results are given in weak-type estimates for almost arbitrary approximation orders, the proofs use only functional analytic methods
The Maximal Riesz Operator of Two-Dimensional Fourier Transforms and Fourier Series on Hp(R×R) and Hp(T×T)

AbstractIt is proved that the maximal operator of the two-parameter Riesz means with parameters α, β⩽1 is bounded from Lp(R2) to Lp(R2) (1<p<∞). The two-dimensional classical Hardy spaces Hp(R×R) are introduced and it is shown that the maximal Riesz operator of a tempered distribution is also bounded from Hp(R×R) to Lp(R2) (max{1/(α+1), 1/(β+1)}<p⩽∞) and is of weak type (H♯1(R×R), L1(R2)) where the Hardy space H♯1(R×R) is defined by the hybrid maximal function. As a consequence we obtain that the Riesz means of a function f∈H♯1(R×R)⊃LlogL(R2) converge a.e. to the function in question. Moreover, we prove that the Riesz means are uniformly bounded on the spaces Hp(R×R) whenever max{1/(α+1), 1/(β+1)}<p<∞. Thus, in case f∈Hp(R×R), the Riesz means converge to f in Hp(R×R) norm. The same results are proved for the conjugate Riesz means and for two-parameter Fourier series, too
Multiple Refinable Hermite Interpolants

AbstractWe consider solutions of a system of refinement equations written in the form as φ(x)=∑n∈Za(n)φ(2x−n),where φ=(φ1, …, φr)T is a vector of compactly supported functions on R and a is a finitely supported sequence of r×r matrices called the refinement mask. If φ is a continuous solution and a is supported on [N1, N2], then v≔(φ(n))N2−1n=N1 is an eigenvector of the matrix (a(2k−n))N2−1k, n=N1 associated with eigenvalue 1. Conversely, given such an eigenvector v, we may ask whether there exists a continuous solution φ such that φ(n)=v(n) for N1⩽n⩽N2−1 (φ(n)=0 for n∉[N1, N2−1], according to the support). The first part of this paper answers this question completely. This existence problem is more general than either the convergence of the subdivision scheme or the requirement of stability, since in one of the latter cases, the eigenvector v is unique up to a constant multiplication. The second part of this paper is concerned with Hermite interpolant solutions, i.e., for some n0∈Z and j, m=1, …, r, φj∈Cr−1(R) and φ(m−1)j(n)=δj, mδn, n0, n∈Z. We provide a necessary and sufficient condition for the refinement equation to have an Hermite interpolant solution. The condition is strictly in terms of the refinement mask. Our method is to characterize the existence and the Hermite interpolant condition by joint spectral radii of matrices. Several concrete examples are presented to illustrate the general theory
On the Parameter Identification Problem in the Plane and the Polar Fractal Interpolation Functions

AbstractFractal interpolation functions provide a new means for fitting experimental data and their graphs can be used to approximate natural scenes. We first determine the conditions that a vertical scaling factor must obey to model effectively an arbitrary function. We then introduce polar fractal interpolation functions as one fractal interpolation method of a non-affine character. Thus, this method may be suitable for a wider range of applications than that of the affine case. The interpolation takes place in polar coordinates and then with an inverse non-affine transformation a simple closed curve arises as an attractor which interpolates the data in the usual plane coordinates. Finally, we prove that this attractor has the same Hausdorff dimension as the polar one
Orthogonal Polynomials on the Circumference and Arcs of the Circumference

AbstractIn this paper we study measures and orthogonal polynomials with asymptotically periodic reflection coefficients. It's known that the support of the orthogonality measure of such polynomials consists of several arcs. We show how the measure of orthogonality can be approximated (resp. described) by the aid of the related orthonormal polynomials if the reflection coefficients are additionally of bounded variation (modN). As an interesting byproduct we obtain that the orthogonality measure is (up to N points) absolutely continuous on the whole circumference, if the reflection coefficients {an} are of bounded variation (modN) and satisfy limn→∞an=0. Furthermore, it is demonstrated that the reflection coefficients remain asymptotically periodic if point measures are added on the support. Finally, we prove that under certain conditions on the arcs orthogonality measures which satisfy a generalized Szego&#x030B; condition have asymptotically periodic reflection coefficients
Continuous Algorithms in n-Term Approximation and Non-Linear Widths

AbstractIn the present paper we investigate optimal continuous algorithms in n-term approximation based on various non-linear n-widths, and n-term approximation by the dictionary V formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with common mixed smoothness. The asymptotic orders of these quantities are given. For each space the asymptotic orders of non-linear n-widths and n-term approximation coincide. Moreover, these asymptotic orders are achieved by a continuous algorithm of n-term approximation by V, which is explicitly constructed
Singular Measures on the Unit Circle and Their Reflection Coefficients

AbstractOrthogonal polynomials on the unit circle are determined by their reflection coefficients through the Szego&#x030B; recurrences. In the present paper we examine two particular classes of measures on the unit circle. The first one consists of measures whose reflection coefficients tend to the unit circle. For such measures we give complete description of their supports (up to the set of isolated masspoints) in terms of reflection coefficients. The supports of measures from the second class have finitely many limit points. We prove the unit circle analogue of M. G. Krein's characterization for the similar class of measures on the real line. The examples of measures from both classes are given
Asymptotics of Kolmogorov Diameters for Some Classes of Harmonic Functions on Spheroids

AbstractLet ΓKD be the unit ball of the space of all bounded harmonic functions in a domain D in R3, considered as a compact subset of the Banach space C(K), where K is a compact subset of D. The old problem about the exact asymptotics for Kolmogorov diameters (widths) of this set,lndk(ΓKD)∼−τk1/2,k→∞, is solved positively in the case when K and D are closed and open confocal spheroids, respectively (i.e., prolate or oblate ellipsoids of revolution). Using some special asymptotic formulas for the associated Legendre functions Pmn(coshσ) as n→∞ and m/n→γ∈[0, 1] (considered earlier by the second author), we show that the constant τ is some averaged characteristic of the pair of spheroids, expressed by means of a certain function of the variable γ, which appears within those asymptotics. Unlike the corresponding problem for analytic functions, quite well investigated, the harmonic functions case has been studied, up to now, only in the case of concentric balls
On Mutually Nearest and Mutually Furthest Points in Reflexive Banach Spaces

AbstractLet G be a nonempty closed (resp. bounded closed) subset in a reflexive strictly convex Kadec Banach space X. Let K(X) denote the space of all nonempty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let KG(X) denote the closure of the set {A∈K(X):A∩G=∅}. A minimization problem min(A, G) (resp. maximization problem max(A, G)) is said to be well posed if it has a unique solution (x0, z0) and every minimizing (resp. maximizing) sequence converges strongly to (x0, z0). We prove that the set of all A∈KG(X) (resp. A∈K(X)) such that the minimization (resp. maximization) problem min(A, G) (resp. max(A, G)) is well posed contains a dense Gδ-subset of KG(X) (resp. K(X)), extending the results in uniformly convex Banach spaces due to Blasi, Myjak and Papini
Dense Subsets of L1-Solutions to Linear Elliptic Partial Differential Equations

AbstractLet Ω⊂RN (N⩾2) be an unbounded domain, and Lm be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L1-solutions to Lm (in Ω) approximate all L1-solutions to Lm (in Ω), provided there exist real numbers Rj→∞, ε⩾0, and a sequence {yj} such that B(yj, ε)∩Ω=∅ and|Λ(yj, Rj, RN\Ω)|RNj>ε ∀ j,where |·| means the volume andΛ(z, R, D)≔∪x∈B(z, R)∩Dz+t(x−z)|x−z|;t⩽1,for z∈RN, R>0 and D⊂RN. For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to Lm may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in C
On the Interpolation by Discrete Splines with Equidistant Nodes

AbstractIn this paper we consider equidistant discrete splines S(j), j∈Z, which may grow as O(|j|s) as |j|→∞. Such splines are relevant for the purposes of digital signal processing. We give the definition of the discrete B-splines and describe their properties. Discrete splines are defined as linear combinations of shifts of the B-splines. We present a solution to the problem of discrete spline cardinal interpolation of the sequences of power growth and prove that the solution is unique within the class of discrete splines of a given order
Best Proximity Pair Theorems for Multifunctions with Open Fibres

AbstractLet A and B be non-empty subsets of a normed linear space, and f:A→B be a single valued function. A solution to the functional equation fx=x, (x∈A) will be an element xo in A such that fxo=xo (i.e., such that d(fx, x)=0). In the case of non-existence of a solution to the equation fx=x, it is natural to explore the existence of an optimal approximate solution that will fulfill the requirement to some extent. In other words, an element xo in A should be found in such a way that d(xo, fxo)=Min{d(x, fx):x∈A}. Thus, the crux of finding an optimal approximate solution to the aforesaid equation fx=x boils down to ascertaining a solution to the optimization problem Min{d(x, fx):x∈A}. But, d(x, fx)⩾d(A, B) for all x∈A. So, in the case of seeking an optimal approximate solution to the aforesaid equation fx=x, it should be contemplated to find an element xo in A such that d(xo, fxo)=d(A, B). Indeed, given a multifunction T: A→2B with open fibres, best proximity pair theorems, furnishing the sufficient conditions for the existence of an element xo∈A such that d(xo, Txo)=d(A, B), are proved in this paper
Approximation by Rectangular Partial Sums of Double Conjugate Fourier Series

AbstractWe consider functions f(x, y) bounded and measurable on the two-dimensional torus T2. The conjugate function f10(x, y) with respect to the first variable is approximated by the rectangular partial sums s10mn(f; x, y) of the corresponding conjugate series as m, n tend to ∞ independently of one another. Our goal is to estimate the rate of this approximation in terms of the oscillation of the function ψ10xy(f; u, v)≔f(x−u, y−v)−f(x+u, y−v)+f(x− u, y+v)−f(x+u, y+v) over appropriate subrectangles of T2. In particular, we obtain a conjugate version of the well-known Dini–Lipschitz test on uniform convergence. We also give estimates in the case where the function f(x, y) is of bounded variation in the sense of Hardy and Krause. Results of similar nature on the one-dimensional torus T were proved in [7]
(C, 1) Means of Orthonormal Expansions for Exponential Weights

AbstractLet sm[f] denote the mth partial sum of the orthonormal expansion of f:R→R with respect to the orthonormal polynomials for the weight W2(x)=exp(−|x|α), α>1. We show that for some C independent of f and n,1n∑m=1nsm[f]Wφ−2/3nL∞(R)⩽C‖fW‖L∞(R) whereφn(x)≔1−xan+n−2/3 and an denotes the nth Mhaskar–Rahmanov–Saff number for Q(x)=12|x|α. The novelty is the presence of the factor φ−2/3n, which is large close to ±an: that factor was absent in the classic results of G. Freud. Related results are proved for more general exponential weights on (−1, 1) or R
Constructive Polynomial Approximation on the Sphere

AbstractThis paper considers the problem of constructive approximation of a continuous function on the unit sphere Sr−1⊆Rr by a spherical polynomial from the space Pn of all spherical polynomials of degree ⩽n. In particular, for r=3 it is shown that the hyperinterpolation approximation Lnf (in which the Fourier coefficients in the exact L2 orthogonal projection Pnf are approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ⩽2n) has the exact order ‖Ln‖≍n1/2 for its uniform norm, provided the underlying quadrature rule satisfies an additional “quadrature regularity” assumption. For r=3, this rate of growth is the same as that of ‖Pn‖, and is known to be optimal among all linear projections on Pn. For r⩾3 an upper bound on ‖Ln‖ of non-optimal asymptotic order O(n(r−1)/2) also holds, without any special assumption on the quadrature rule
Entropy of C(K)-Valued Operators

AbstractWe investigate how the entropy numbers (en(T)) of an arbitrary Hölder-continuous operator T:E→C(K) are influenced by the entropy numbers (εn(K)) of the underlying compact metric space K and the geometry of E. We derive diverse universal inequalities relating finitely many εn(K)'s with finitely many en(T)'s which yield statements about the asymptotically optimal behaviour of the sequence (en(T)) in terms of the sequence (εn(K)). As an application we present new methods for estimating the entropy numbers of a precompact and convex subset in a Banach space E, provided that the entropy numbers of its extremal points are known
The Density of Translates of Zonal Kernels on Compact Homogeneous Spaces

AbstractCompact manifolds embedded in Euclidean space which have a transitive group G of linear isometries, such as the spheres with the rotation group or the “flat” tori with the group of rotations in each coordinate direction, admit a natural notion of a continuous G-invariant kernel function k(x, y), which generalizes the idea of a radial or distance-dependent function on the spheres and tori. In connection with a study of quasi-interpolation on these spaces, we have reproved and extended results of Sun for the spheres to characterize those kernels for which the span of the translates, ∑ank(x, yn), is dense in the continuous functions. The essence of the characterization is that the integral operator with G-invariant kernel k(x, y) must be non-singular when restricted to the space of nth degree polynomial functions. This requires that the polynomials be invariant under all such linear operators, which is true for many compact homogeneous M including the spheres, tori, and others. In fact the non-singularity must hold only on any finite-dimensional space of zonal polynomials, those which are pointwise fixed by the subgroup of all isometries fixing a single point on M. In practical terms this later condition is verified by choosing one point on the manifold (the north pole on the spheres or the identity element on the flat tori), picking some basis for the polynomials of given degree which are fixed under the isometries leaving the pole invariant, and testing whether the integral operator (which leaves this space invariant) has a non-singular matrix. In all the cases considered, where the family of G-invariant kernels lead to commuting operator families, there are diagonalizing bases for this restricted operator, and the characterization becomes the non-vanishing of the appropriate Fourier-like coefficients
Wavelet Approximation of Periodic Functions

AbstractWe investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and Lp norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough
Complexity of Weighted Approximation over R

AbstractWe study approximation of univariate functions defined over the reals. We assume that the rth derivative of a function is bounded in a weighted Lp norm with a weight ψ. Approximation algorithms use the values of a function and its derivatives up to order r−1. The worst case error of an algorithm is defined in a weighted Lq norm with a weight ρ. We study the worst case (information) complexity of the weighted approximation problem, which is equal to the minimal number of function and derivative evaluations needed to obtain error ε. We provide necessary and sufficient conditions in terms of the weights ψ and ρ, as well as the parameters r, p, and q for the weighted approximation problem to have finite complexity. We also provide conditions which guarantee that the complexity of weighted approximation is of the same order as the complexity of the classical approximation problem over a finite interval. Such necessary and sufficient conditions are also provided for a weighted integration problem since its complexity is equivalent to the complexity of the weighted approximation problem for q=1
Convergence of Cascade Algorithms in Sobolev Spaces Associated with Inhomogeneous Refinement Equations

AbstractIn this paper, we consider multivariate inhomogeneous refinement equations of the form ϕ(x)=∑α∈Zsa(α)ϕ(2x−α)+g(x), x∈Rs, where ϕ=(ϕ1, …, ϕr)T is the unknown, g=(g1, …, gr)T is a given vector of functions on Rs, and a is a finitely supported refinement mask such that each a(α) is an r×r (complex) matrix. Let ϕ0 be an initial vector of functions in the Sobolev space Wk2(Rs). The corresponding cascade algorithm is given by ϕn(x)=∑α∈Zsa(α)ϕn−1(2x−α)+g(x), x∈Rs, n=1, 2, …. A characterization is given for the strong convergence of the cascade algorithm in the Sobolev space Wk2(Rs) (k∈N) in terms of the refinement mask a, the inhomogeneous term g, and the initial vector of functions ϕ0