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Wavelet characterization of exponentially weighted Besov space with dominating mixed smoothness and its application to function approximation J. Approx. Theory (IF 0.9) Pub Date : 2024-03-11 Yoshihiro Kogure, Ken’ichiro Tanaka
Although numerous studies have focused on normal Besov spaces, limited studies have been conducted on exponentially weighted Besov spaces. Therefore, we define exponentially weighted Besov space whose smoothness includes normal Besov spaces, Besov spaces with dominating mixed smoothness, and their interpolation. Furthermore, we obtain wavelet characterization of . Next, approximation formulas such
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Infinite-dimensional integration and L2-approximation on Hermite spaces J. Approx. Theory (IF 0.9) Pub Date : 2024-02-08 M. Gnewuch, A. Hinrichs, K. Ritter, R. Rüßmann
We study integration and -approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of
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Inradius of random lemniscates J. Approx. Theory (IF 0.9) Pub Date : 2024-02-03 Manjunath Krishnapur, Erik Lundberg, Koushik Ramachandran
A classically studied geometric property associated to a complex polynomial is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate .
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Estimates of linear expressions through factorization J. Approx. Theory (IF 0.9) Pub Date : 2024-01-23 Ali Hasan Ali, Zsolt Páles
The aim of this paper is to establish various factorization results and then to derive estimates for linear functionals through the use of a generalized Taylor theorem. Additionally, several error bounds are established including applications to the trapezoidal rule as well as to a Simpson formula-type rule.
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Multivariate polynomial splines on generalized oranges J. Approx. Theory (IF 0.9) Pub Date : 2024-01-15 Maritza Sirvent, Tatyana Sorokina, Nelly Villamizar, Beihui Yuan
We consider spaces of multivariate splines defined on a particular type of simplicial partitions that we call . Such partitions are composed of a finite number of maximal faces with exactly one shared face. We reduce the problem of finding the dimension of splines on oranges to computing dimensions of splines on simpler, lower-dimensional partitions that we call . We use both algebraic and Bernstein–Bézier tools
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Localization for random CMV matrices J. Approx. Theory (IF 0.9) Pub Date : 2024-01-05 Xiaowen Zhu
We prove Anderson localization (AL) and dynamical localization in expectation (EDL, also known as strong dynamical localization) for random CMV matrices for arbitrary distribution of i.i.d. Verblunsky coefficients.
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On some identities for confluent hypergeometric functions and Bessel functions J. Approx. Theory (IF 0.9) Pub Date : 2024-01-03 Yoshitaka Okuyama
Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula
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Polynomial approximation on disjoint segments and amplification of approximation J. Approx. Theory (IF 0.9) Pub Date : 2024-01-05 Yu. Malykhin, K. Ryutin
We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments (see (1)). This problem has important applications in several areas of numerical analysis, complexity theory, quantum algorithms, etc. The one, most relevant for us, is the amplification of approximation method: it allows to construct approximations
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Comonotone approximation of periodic functions J. Approx. Theory (IF 0.9) Pub Date : 2024-01-05 D. Leviatan, M.V. Shchehlov, I.O. Shevchuk
Let be the space of continuous -periodic functions , endowed with the uniform norm , and denote by , the th modulus of smoothness of . Denote by , the subspace of times continuously differentiable functions , and let , be the set of trigonometric polynomials of degree . If , has , , extremal points in , denote by the error of its best comonotone approximation. We prove, that if , then for either ,
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Chebyshev unions of planes, and their approximative and geometric properties J. Approx. Theory (IF 0.9) Pub Date : 2023-12-30 A.R. Alimov, I.G. Tsar’kov
We study approximative and geometric properties of Chebyshev sets composed of at most countably many planes (i.e., closed affine subspaces). We will assume that the union of planes is irreducible, i.e., no plane in this union contains another plane from the union. We show, in particular, that if a Chebyshev subset M of a Banach space X consists of at least two planes, then it is not B-connected (i
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Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints J. Approx. Theory (IF 0.9) Pub Date : 2024-01-04 German Dzyubenko, Kirill A. Kopotun
Given , a nonnegative function , , an arbitrary finite collection of points , and a corresponding collection of nonnegative integers with , , is it true that, for sufficiently large , there exists a polynomial of degree such that
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Is hyperinterpolation efficient in the approximation of singular and oscillatory functions? J. Approx. Theory (IF 0.9) Pub Date : 2024-01-04 Congpei An, Hao-Ning Wu
Singular and oscillatory functions play a crucial role in various applications, and their approximation is crucial for solving applied mathematics problems efficiently. Hyperinterpolation is a discrete projection method approximating functions with the orthogonal projection coefficients obtained by numerical integration. However, this approach may be inefficient for approximating singular and oscillatory
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Onesided Korovkin approximation J. Approx. Theory (IF 0.9) Pub Date : 2024-01-03 Michele Campiti
In this paper we study in detail some characterizations of Korovkin closures and we also introduce the notions of onesided upper and lower Korovkin closures. We provide some complete characterizations of these new closures which separate the roles of approximating functions in a Korovkin system. We also present some new characterizations of the classical Korovkin closure in spaces of integrable functions
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An asymptotic development of the Poisson integral for Laguerre polynomial expansions J. Approx. Theory (IF 0.9) Pub Date : 2023-12-02 Ulrich Abel
The purpose of this paper is the study of the rate of convergence of Poisson integrals for Laguerre expansions. The convergence of partial sums of Fourier series of functions in Lp spaces was studied, for several classes of orthogonal polynomials. In the Laguerre case Askey and Waigner proved convergence for functions f∈Lp0,+∞ with 4/3
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Littlewood–Paley–Rubio de Francia inequality for unbounded Vilenkin systems J. Approx. Theory (IF 0.9) Pub Date : 2023-11-29 Anton Tselishchev
Rubio de Francia proved the one-sided version of Littlewood–Paley inequality for arbitrary intervals. In this paper, we prove the similar inequality in the context of arbitrary Vilenkin systems (that is, for functions on infinite products of cyclic groups). There are no assumptions on the orders of these groups.
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The sharp Landau–Kolmogorov inequality for the set ‖y′‖2, ‖y‖1, ‖y+′′‖∞ on the real line J. Approx. Theory (IF 0.9) Pub Date : 2023-11-21 N.S. Payuchenko
We obtain the sharp Kolmogorov inequality ‖y′‖L2(G)≤223‖y‖L1(G)1/2‖y+′′‖L∞(G)1/2 on the real line G=R and the period G=[0,1).
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Coefficient-based regularized distribution regression J. Approx. Theory (IF 0.9) Pub Date : 2023-11-04 Yuan Mao, Lei Shi, Zheng-Chu Guo
In this paper, we consider the coefficient-based regularized distribution regression which aims to regress from probability measures to real-valued responses over a reproducing kernel Hilbert space (RKHS), where the regularization is put on the coefficients and kernels are assumed to be indefinite. The algorithm involves two stages of sampling, the first stage sample consists of distributions and the
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Triebel–Lizorkin regularity and bi-Lipschitz maps: Composition operator and inverse function regularity J. Approx. Theory (IF 0.9) Pub Date : 2023-10-18 Martí Prats
We study the stability of Triebel–Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel–Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain the results we provide an equivalent norm for the Triebel–Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order
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Mean convergence of Fourier–Akhiezer–Chebyshev series J. Approx. Theory (IF 0.9) Pub Date : 2023-10-11 Manuel Bello-Hernández, Alejandro del Campo López
We prove mean convergence of the Fourier series in Akhiezer-Chebyshev polynomials in Lp, p>1, using a weighted inequality for the Hilbert transform in an arc of the unit circle.
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Richard S. Varga October 9, 1928 – February 25, 2022 J. Approx. Theory (IF 0.9) Pub Date : 2023-10-05 Vladimir Andrievskii, András Kroó, József Szabados
Abstract not available
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Approximating properties of metric and generalized metric projections in uniformly convex and uniformly smooth Banach spaces J. Approx. Theory (IF 0.9) Pub Date : 2023-09-27 Akhtar A. Khan, Jinlu Li
This note conducts a comparative study of some approximating properties of the metric projection, generalized projection, and generalized metric projection in uniformly convex and uniformly smooth Banach spaces. We prove that the inverse images of the metric projections are closed and convex cones, but they are not necessarily convex. In contrast, inverse images of the generalized projection are closed
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Analytical study of the pantograph equation using Jacobi theta functions J. Approx. Theory (IF 0.9) Pub Date : 2023-09-24 Changgui Zhang
The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y′(x)=ay(qx)+by(x), where a and b are two non-zero real or complex numbers. When 0
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Orthogonal polynomials in weighted Bergman spaces J. Approx. Theory (IF 0.9) Pub Date : 2023-09-24 Erwin Miña-Díaz
Let w be a weight on the unit disk D having the form w(z)=|v(z)|2∏k=1sz−ak1−za¯kmk,mk>−2,|ak|<1,where v is analytic and free of zeros in D¯, and let (pn)n=0∞ be the sequence of polynomials (pn of degree n) orthonormal over D with respect to w. We give an integral representation for pn from which it is in principle possible to derive its asymptotic behavior as n→∞ at every point z of the complex plane
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Counterexamples in isometric theory of symmetric and greedy bases J. Approx. Theory (IF 0.9) Pub Date : 2023-09-22 Fernando Albiac, José L. Ansorena, Óscar Blasco, Hùng Việt Chu, Timur Oikhberg
We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017)
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The upper bound for the Lebesgue constant for Lagrange interpolation in equally spaced points of the triangle J. Approx. Theory (IF 0.9) Pub Date : 2023-09-11 Natalia Baidakova
An upper bound for the Lebesgue constant, i.e., the supremum norm of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to n is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to n for an arbitrary d-dimensional simplex was established by the author. The upper bound proved in this article
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Quarklet characterizations for Triebel–Lizorkin spaces J. Approx. Theory (IF 0.9) Pub Date : 2023-09-09 Marc Hovemann, Stephan Dahlke
In this paper we prove that under some conditions on the parameters the univariate Triebel–Lizorkin spaces Fr,qs(R) can be characterized in terms of quarklets. So for functions from Triebel–Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed by means of biorthogonal compactly supported Cohen–Daubechies–Feauveau
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Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces J. Approx. Theory (IF 0.9) Pub Date : 2023-08-22 Morten Nielsen
We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on Rd. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous
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On semi-classical weight functions on the unit circle J. Approx. Theory (IF 0.9) Pub Date : 2023-08-16 Cleonice F. Bracciali, Karina S. Rampazzi, Luana L. Silva Ribeiro
We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. We determine all such semi-classical weight functions and this also includes an extension of the Jacobi weight function on the unit circle. General structure
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On dynamics of asymptotically minimal polynomials J. Approx. Theory (IF 0.9) Pub Date : 2023-07-29 Turgay Bayraktar, Melİke Efe
We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are sufficiently close to E then we show that the filled Julia
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A modified Christoffel function and its asymptotic properties J. Approx. Theory (IF 0.9) Pub Date : 2023-07-29 Jean B. Lasserre
We introduce a certain variant (or regularization) Λ̃nμ of the standard Christoffel function Λnμ associated with a measure μ on a compact set Ω⊂Rd. Its reciprocal is now a sum-of-squares polynomial in the variables (x,ɛ), ɛ>0. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with n of its inverse is at most polynomial inside and exponential outside the
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Harmonic analysis of little q-Legendre polynomials J. Approx. Theory (IF 0.9) Pub Date : 2023-07-17 Stefan Kahler
Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials. The individual behavior
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Explicit expressions and computational methods for the Fortet-Mourier distance of positive measures to finite weighted sums of Dirac measures J. Approx. Theory (IF 0.9) Pub Date : 2023-07-16
Explicit expressions and computational approaches are given for the Fortet-Mourier distance between a positively weighted sum of Dirac measures on a metric space and a positive finite Borel measure. Explicit expressions are given for the distance to a single Dirac measure. For the case of a sum of several Dirac measures one needs to resort to a computational approach. In particular, two algorithms
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Proximinality and uniformly approximable sets in Lp J. Approx. Theory (IF 0.9) Pub Date : 2023-07-11 Guillaume Grelier, Jaime San Martín
For any p∈[1,∞], we prove that the set of simple functions taking at most k different values is proximinal in Lp for all k≥1. Moreover, if 1
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Approximation error for neural network operators by an averaged modulus of smoothness J. Approx. Theory (IF 0.9) Pub Date : 2023-07-07 Danilo Costarelli
In the present paper we establish estimates for the error of approximation (in the Lp-norm) achieved by neural network (NN) operators. The above estimates have been given by means of an averaged modulus of smoothness introduced by Sendov and Popov, also known with the name of τ-modulus, in case of bounded and measurable functions on the interval [−1,1]. As a consequence of the above estimates, we can
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Telescoping continued fractions for the error term in Stirling’s formula J. Approx. Theory (IF 0.9) Pub Date : 2023-07-04 Gaurav Bhatnagar, Krishnan Rajkumar
In this paper, we introduce telescoping continued fractions to find lower bounds for the error term rn in Stirling’s approximation n!=2πnn+1/2e−nern. This improves lower bounds given earlier by Cesàro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction
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H-sets for kernel-based spaces J. Approx. Theory (IF 0.9) Pub Date : 2023-06-26 Robert Schaback
The concept of H-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and H-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, H-sets are shown here to have a rather simple
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Discrete harmonic analysis associated with Jacobi expansions III: The Littlewood–Paley–Stein gk-functions and the Laplace type multipliers J. Approx. Theory (IF 0.9) Pub Date : 2023-06-26 Alberto Arenas, Óscar Ciaurri, Edgar Labarga
The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator J(α,β)=J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials and I is the identity operator, we define the corresponding Littlewood–Paley–Stein gk(α,β)-functions associated with
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Sharp Lp-error estimates for sampling operators J. Approx. Theory (IF 0.9) Pub Date : 2023-06-26 Yurii Kolomoitsev, Tetiana Lomako
We study approximation properties of linear sampling operators in the spaces Lp for 1≤p<∞. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in Lp and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical
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Weighted Lp Markov factors with doubling weights on the ball J. Approx. Theory (IF 0.9) Pub Date : 2023-06-25 Jiansong Li, Heping Wang, Kai Wang
Let Lp,w,1≤p<∞, denote the weighted Lp space of functions on the unit ball Bd with a doubling weight w on Bd. The Markov factor for Lp,w of a polynomial P is defined by ‖|∇P|‖p,w‖P‖p,w, where ∇P is the gradient of P. We investigate the worst case Markov factors for Lp,w and prove that the degree of these factors is at most 2. In particular, for the Gegenbauer weight wμ(x)=(1−|x|2)μ−1/2,μ≥0, the exponent
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Quasi-uniform designs with optimal and near-optimal uniformity constant J. Approx. Theory (IF 0.9) Pub Date : 2023-06-14 L. Pronzato, A. Zhigljavsky
A design is a collection of distinct points in a given set X, which is assumed to be a compact subset of Rd, and the mesh-ratio of a design is the ratio of its fill distance to its separation radius. The uniformity constant of a sequence of nested designs is the smallest upper bound for the mesh-ratios of the designs. We derive a lower bound on this uniformity constant and show that a simple greedy
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Absolute minima of potentials of certain regular spherical configurations J. Approx. Theory (IF 0.9) Pub Date : 2023-06-08 Sergiy Borodachov
We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical (2m−3)-designs with a non-trivial index 2m that are contained in a union of m parallel hyperplanes, m≥2, whose locations satisfy certain additional assumptions. The interaction between points is described by a function of the dot product, which has positive derivatives of orders 2m−2, 2m−1
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On a conjecture concerning interpolation by bivariate Bernstein polynomials J. Approx. Theory (IF 0.9) Pub Date : 2023-06-01 Michael S. Floater
In this paper we discuss a conjecture of Schumaker that the principal submatrices of collocation matrices of bivariate Bernstein polynomials over triangular grids have positive determinant. It is easy to show that the conjecture holds for the 2 × 2 submatrices. In this paper we show that it also holds for the 3 × 3 submatrices, working with the equivalent ‘monomial form’ of the conjecture. This result
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Random sections of ℓp-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators J. Approx. Theory (IF 0.9) Pub Date : 2023-06-01 Aicke Hinrichs, Joscha Prochno, Mathias Sonnleitner
We study the circumradius of a random section of an ℓp-ellipsoid, 0
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Sharp estimates for Jacobi heat kernels in conic domains J. Approx. Theory (IF 0.9) Pub Date : 2023-05-29 Dawid Hanrahan, Dariusz Kosz
We prove genuinely sharp estimates for the Jacobi heat kernels introduced in the context of the multidimensional cone Vd+1 and its surface V0d+1. To do so, we combine the theory of Jacobi polynomials on the cone explored by Xu with the recent techniques by Nowak, Sjögren, and Szarek, developed to find genuinely sharp estimates for the spherical heat kernel.
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Strong uniqueness and alternation theorems for relative Chebyshev centers J. Approx. Theory (IF 0.9) Pub Date : 2023-05-27 F.E. Levis, C.V. Ridolfi, L. Zabala
In this paper, we give a strong uniqueness characterization theorem for the Chebyshev center of a set of infinitely many functions relative to a finite-dimensional linear space on a compact Hausdorff space. Additionally, we derive an alternation theorem for Chebyshev centers relative to a weak Chebyshev space on any compact set of the real line. Furthermore, we show an intrinsic characterization of
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Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials J. Approx. Theory (IF 0.9) Pub Date : 2023-05-27 Diego Dominici, Juan José Moreno-Balcázar
In this paper we tackle the asymptotic behavior of a family of orthogonal polynomials with respect to a nonstandard inner product involving the forward operator Δ. Concretely, we treat the generalized Charlier weights in the framework of Δ-Sobolev orthogonality. We obtain an asymptotic expansion for these orthogonal polynomials where the falling factorial polynomials play an important role.
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Sampling discretization error of integral norms for function classes with small smoothness J. Approx. Theory (IF 0.9) Pub Date : 2023-05-22 V.N. Temlyakov
We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the square norm on different function
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The norm of the Cesàro operator minus the identity and related operators acting on decreasing sequences J. Approx. Theory (IF 0.9) Pub Date : 2023-05-08 Santiago Boza, Javier Soria
Recently, several authors have considered the problem of determining optimal norm inequalities for discrete Hardy-type operators (like Cesàro or Copson). In this work, we obtain sharp bounds for the norms of the difference of the Cesàro operator with either the identity or the shift, when they are restricted to the cone of decreasing sequences in ℓp (which is closely related to the previously mentioned
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Boundary value problems of potential theory for the exterior ball and the approximation and ergodic behaviour of the solutions J. Approx. Theory (IF 0.9) Pub Date : 2023-05-10 P.L. Butzer, R.L. Stens
The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in R3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role. The solutions themselves are evaluated by means of Fourier expansions with
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Mean convergence of Lagrange interpolation J. Approx. Theory (IF 0.9) Pub Date : 2023-05-09 Glenier Bello, Manuel Bello-Hernández
In this note we prove mean convergence of Lagrange interpolation at the zeros of para-orthogonal polynomials for measures on the unit circle which do not belong to Szegő’s class. When the measure is in Szegő’s class, mean convergence of Lagrange interpolation is proved for functions in the disk algebra.
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Spectral properties of a class of Moran measures on R2 J. Approx. Theory (IF 0.9) Pub Date : 2023-05-09 Zhi-Hui Yan
Given a pair (R,D), where R={Ri}i=1∞ is a sequence of expanding matrix (i.e., all the eigenvalues of Ri have modulus strictly greater than 1), and D={Di}i=1∞⊆Z2. It is well known that there exists an infinite convolution generated by (R,D) which satisfies μR,D≔δR1−1D1∗δ(R2R1)−1D2∗⋯,we say that μR,D is a Moran measure if it convergent to a probability measure with compact support in a weak sense, where
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Hermite–Padé approximation and integrability J. Approx. Theory (IF 0.9) Pub Date : 2023-05-09 Adam Doliwa, Artur Siemaszko
We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearance of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other
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Approximation by sums of shifts and dilations of a single function and neural networks J. Approx. Theory (IF 0.9) Pub Date : 2023-05-08 K. Shklyaev
We find sufficient conditions on a function f to ensure that sums of functions of the form f(αx−θ), where α∈A⊂R and θ∈Θ⊂R, are dense in the real spaces C0 and Lp on the real line or its compact subsets. That is, we consider linear combinations in which all coefficients are 1. As a corollary we deduce results on density of sums of functions f(w⋅x−θ), w∈W⊂Rd, θ∈Θ⊂R in C(Rd) in the topology of uniform
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Approximation by linear combinations of translates in invariant Banach spaces of tempered distributions via Tauberian conditions J. Approx. Theory (IF 0.9) Pub Date : 2023-05-02 Hans G. Feichtinger, Anupam Gumber
This paper describes an approximation theoretic approach to the problem of completeness of a set of translates of a “Tauberian generator”, which is an integrable function whose Fourier transform does not vanish. This is achieved by the construction of finite rank operators, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the
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Estimates of Lebesgue constants for Lagrange interpolation processes by rational functions under mild restrictions to their fixed poles J. Approx. Theory (IF 0.9) Pub Date : 2023-04-28 Sergei Kalmykov, Alexey Lukashov
We estimate the Lebesgue constants for Lagrange interpolation processes on one or several intervals by rational functions with fixed poles. We admit that the poles have finitely many accumulation points on the intervals. To prove it we use an analog of the inverse polynomial image method for rational functions with fixed poles.
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Bernstein-type constants for approximation of |x|α by partial Fourier–Legendre and Fourier–Chebyshev sums J. Approx. Theory (IF 0.9) Pub Date : 2023-04-17 Wenjie Liu, Li-Lian Wang, Boying Wu
In this paper, we study the approximation of fα(x)=|x|α,α>0 in L∞[−1,1] by its Fourier–Legendre partial sum Sn(α)(x). We derive the upper and lower bounds of the approximation error in the L∞-norm that are valid uniformly for all n≥n0 for some n0≥1. Such an optimal L∞-estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can
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Density results and trace operator in weighted Sobolev spaces defined on the half-line, equipped with power weights J. Approx. Theory (IF 0.9) Pub Date : 2023-04-12 Radosław Kaczmarek, Agnieszka Kałamajska
We study properties of W01,p(R+,tβ) — the completion of C0∞(R+) in the power-weighted Sobolev spaces W1,p(R+,tβ), where β∈R. Among other results, we obtain the analytic characterization of W01,p(R+,tβ) for all β∈R. Our analysis is based on the precise study of the two trace operators: Tr0(u)≔limt→0u(t) and Tr∞(u)≔limt→∞u(t), which leads to the analysis of the asymptotic behavior of functions from W01
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Bernstein inequality on conic domains and triangles J. Approx. Theory (IF 0.9) Pub Date : 2023-03-16 Yuan Xu
We establish weighted Bernstein inequalities in Lp space for the doubling weight on the conic surface V0d+1={(x,t):‖x‖=t,x∈Rd,t∈[0,1]} as well as on the solid cone bounded by the conic surface and the hyperplane t=1, which becomes a triangle on the plane when d=1. While the inequalities for the derivatives in the t variable behave as expected, there are inequalities for the derivatives in the x variables