In this paper we study approximation theorems for L2-space on Damek–Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek–Ricci spaces. We also prove Bernstein–Nikolskii–Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application we

The main result of this paper is an intersection representation for a class of anisotropic vector-valued function spaces in an axiomatic setting à la Hedberg and Netrusov (2007), which includes weighted anisotropic mixed-norm Besov and Lizorkin-Triebel spaces. In the special case of the classical Lizorkin-Triebel spaces, the intersection representation gives an improvement of the well-known Fubini

We study a dual-type problem to generalized Christoffel function. The solution is connected with other extremal problems in the Hp space of analytic functions on the unit circle considered by Macintyre, Rogosinski and Shapiro.

We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that any algebraic polynomial of degree n approximating a function f∈Cr(I), I=[−1,1], at the classical pointwise rate c(k,r)ρnr(x)ωk(f(r)

Let Xρ be a modulated convergence space, that is, a modular space equipped with a sequential convergence structure. Given an element x of Xρ, we consider the minimisation problem of finding x0∈C such that ρ(x−xo)=inf{ρ(x−y):y∈C}, where ρ is a convex modular and C is a closed convex subset of Xρ. Such an element x0 is called a best approximant. We prove existence and uniqueness of such a best approximant

In this paper, we study the distributed learning algorithm and the distribution regression problem of coefficient regularization for Mercer kernels. By utilizing divided-and-conquer approach, we partition a data set into disjoint data subsets for different learning machines, and get the global estimator from local estimators. By using second order decomposition on the difference of operator inverse

Information theoretic learning is a learning paradigm that uses concepts of entropies and divergences from information theory. A variety of signal processing and machine learning methods fall into this framework. Minimum error entropy principle is a typical one amongst them. In this paper, we study a kernel version of minimum error entropy methods that can be used to find nonlinear structures in the

Meixner type polynomials (qn)n≥0 are defined from the Meixner polynomials by using Casoratian determinants whose entries belong to two given finite sets of polynomials (Sh)h=1m1 and (Tg)g=1m2. They are eigenfunctions of higher order difference operators but only for a careful choice of the polynomials (Sh)h=1m1 and (Tg)g=1m2, the sequence (qn)n≥0 is orthogonal with respect to a measure. In this paper

In a larger size Riemann–Hilbert problem matching the local parametrices with the global parametrix is often a major technical issue. In this article we present a result that should tackle this problem in natural situations. We prove that, in a general setting, it is possible to obtain a double matching, that is, a matching condition on two circles instead of one circle. We discuss how this matching

In this paper, we discuss the compressed data separation problem. In order to reconstruct the distinct subcomponents, which are sparse in morphologically different dictionaries D1∈Rn×d1 and D2∈Rn×d2, we present a general class of convex optimization decoder. It can deal with signal separation under the corruption of different kinds of noises, including Gaussian noise (p=2), Laplacian noise (p=1), and

We consider special families of orthogonal polynomials satisfying differential equations. Besides known hypergeometric cases, we look especially for Heun’s differential equations. We show that such equations are satisfied by orthogonal polynomials related to some classical weight functions modified by Dirac weights or by division of powers of binomials. An appropriate set of biorthogonal rational functions

We study weighted Besov and Triebel–Lizorkin spaces associated with Hermite expansions and obtain (i) frame decompositions, and (ii) characterizations of continuous Sobolev-type embeddings. The weights we consider generalize the Muckhenhoupt weights.

We study an inverse problem that consists in estimating the first (zero-order) moment of some R2-valued distribution m that is supported within a closed interval S̄⊂R, from partial knowledge of the solution to the Poisson-Laplace partial differential equation with source term equal to the divergence of m on another interval parallel to and located at some distance from S. Such a question coincides

Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ingredient for this to be successful is the distribution of the sampling points. Since the computation of an optimal selection of sampling points may be an infeasible task, one promising option is to use greedy methods. Although

The main aim of this article is a careful investigation of the asymptotic behavior of zeros of Bernoulli polynomials of the second kind. It is shown that the zeros are all real and simple. The asymptotic expansions for the small, large, and the middle zeros are computed in more detail. The analysis is based on the asymptotic expansions of the Bernoulli polynomials of the second kind in various regimes

In this paper, we find strong uniqueness constants for a certain class of operators with a unit norm from a space of dimension 2n onto its subspace of codimension n, which is formed by using hyperplanes in l∞2n(n≥2).

Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials Pn(z) defined by Jacobi recurrence coefficients an (off-diagonal terms) and bn (diagonal terms). We consider the case an→∞, bn→∞ in such a way that ∑an−1<∞ (that is, the Carleman condition is violated) and γn:=2−1bn(anan−1)−1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for Pn(z) are known; they depend crucially on

We study approximation properties of weighted L2-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form (1−‖x‖2)α∏i=1d|xi|γi, α,γ1,…,γd>−1. Said properties are measured in Dunkl–Sobolev-type norms in which the same weighted L2 norm is used to control all the involved differential–difference Dunkl operators

In this paper we will study the harmonic analysis of a class of Cantor–Moran measures μ with three-element digit sets on R. Our results give some sufficient conditions for the maximal orthonormal set of exponential functions to be or not to be a basis for the space L2(μ).

We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when 0 lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when the orthogonality measure is absolutely continuous and we provide a constructive formula for it in terms of the limit of Turán determinants. We next consider asymptotics of the solutions of associated second

We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even

The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg

Let L1,L2,…,LK be a family of closed subspaces of a Hilbert space H, L1∩⋯∩LK={0}; let Pk be the orthogonal projection onto Lk. We consider two types of consecutive projections of an element x0∈H: alternating projections Tnx0, where T=PK∘⋯∘P1, and remotest projections xn defined recursively, xn+1 being the remotest point for xn among P1xn,…,PKxn. These xn can be interpreted as residuals in greedy approximation

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives

In this paper, we study the sequence of orthogonal polynomials {Sn}n=0∞ with respect to the Sobolev-type inner product 〈f,g〉=∫−11f(x)g(x)dμ(x)+∑j=1Nηjf(dj)(cj)g(dj)(cj)where μ is a finite positive Borel measure whose support suppμ⊂[−1,1] contains an infinite set of points, ηj>0, N,dj∈Z+ and {c1,…,cN}⊂R∖[−1,1]. Under some restriction of order in the discrete part of 〈⋅,⋅〉, we prove that for sufficiently

We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from Breuer (2011) showing that clock spacing holds even for certain singular continuous measures.

Particular class of skew orthogonal polynomials are introduced and investigated, which possess Laurent symmetry. They are also shown to appear as eigenfunctions of symplectic generalized eigenvalue problems. Furthermore, the modification of these polynomials gives some symplectic eigenvalue problem and the corresponding symplectic matrix is equivalent to butterfly matrix, which is a canonical form

We continue our study of the Widom factors for Lp(μ) extremal polynomials initiated in (Alpan and Zinchenko, 2020). In this work we characterize sets for which the lower bounds obtained in (Alpan and Zinchenko, 2020) are saturated, establish continuity of the Widom factors with respect to the measure μ, and show that despite the lower bound [W2,n(μK)]2≥2S(μK) for the equilibrium measure μK on a compact

Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual function from a given function class its sequence of errors of best sparse approximations with respect to a given dictionary decays faster than the corresponding

Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data’s structure or on the behavior of the unknown function can make the task well-posed. In this work, we study

We state and prove three general formulas allowing us to transform formal finite sums into formal continued fractions and use them to generalize certain expansions in regular continued fractions given by Hone and Varona. As an application, we obtain formulas of transformation of certain series into regular continued fractions. For example, we exhibit a sequence (xn) of positive integers which satisfies

Let S(Rn) be the Schwartz class on Rn and S∞(Rn)≔ϕ∈S(Rn):∫Rnxαϕ(x)dx=0for any multi-indexα∈({0,1,…})n,and S′(Rn) and S∞′(Rn) be their dual spaces, respectively. Let a→≔(a1,…,an)∈[1,∞)n, p→≔(p1,…,pn)∈(0,1]n, and Ha→p→(Rn)⊂S′(Rn) be the anisotropic mixed-norm Hardy space, associated with an anisotropic quasi-homogeneous norm |⋅|a→, defined via the non-tangential grand maximal function. In this article

In this paper we study direct and inverse approximation inequalities in Lp(Rd), 1

We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in terms of certain continued fractions such that this stochastic factorization is always possible. By inverting the order of the factors (also known as

Let {ηj}j=0N be a sequence of independent and identically distributed random complex Gaussian variables, and let {fj(z)}j=0N be a sequence of given analytic functions that are real-valued on the real line. We prove an exact formula for the expected density of the distribution of complex zeros of the random equation ∑j=0Nηjfj(z)=K, where K∈C. The method of proof employs a formula for the expected absolute

This paper is devoted to the decomposition of vectors into sampled complex exponentials; or, equivalently, to the information over discrete measures captured in a finite sequence of their Fourier coefficients. We study existence, uniqueness, and cardinality properties, as well as computational aspects of estimation using convex semidefinite programs. We then explore optimal transport between measures

Suppose P≔(pj)j=1∞ is a sequence of distinct real numbers pj>0. We prove that the equalities ‖f‖p=‖g‖p,p∈P, imply μ({x∈E:|f(x)|<α})=μ({x∈E:|g(x)|<α}),α≥0, whenever 0<μ(E)<∞ and f,g∈L∞(E) if and only if ∑j=1∞pjpj2+1=∞.

The unit ball Bpn(R) of the finite-dimensional Schatten trace class Spn consists of all real n×n matrices A whose singular values s1(A),…,sn(A) satisfy s1p(A)+…+snp(A)≤1, where p>0. Saint Raymond (1984) showed that the limit limn→∞n1∕2+1∕p(VolBpn(R))1∕n2 exists in (0,∞) and provided both lower and upper bounds. In this manuscript we use the theory of logarithmic potentials in external fields to determine

Given a sequence of Marcinkiewicz-Zygmund inequalities in L2, we derive approximation theorems and quadrature rules. The derivation is completely elementary and requires only the definition of Marcinkiewicz-Zygmund inequality, Sobolev spaces, and the solution of least square problems.

After the appearance of the article “A retrospective on 60 years of approximation theory and associated fields” (J. Approx. Theory 160 (1–2) (2009) 3–18), several readers informed me (PLB) that they would like to see an article based upon research contacts and conference participations of members of the chair “Lehrstuhl A für Mathematik” at Aachen throughout the world. The present paper is devoted

This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real

We correct a mistake in the statement of Theorem 2.4.

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because

We show that if U is an arbitrary open subset of a Riemann surface and φ an arbitrary continuous function on the boundary ∂U, then there exists a holomorphic function φ˜ on U such that, for every p∈∂U, φ˜(x)→φ(p), as x→p outside a set of density 0 at p relative to U. These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume

In this paper we prove that if a multivariate function of a certain smoothness class is represented by a sum of k arbitrarily behaved ridge functions, then it can be represented by a sum of k ridge functions of the same smoothness class and a polynomial of degree at most k−1. This solves the problem posed by A. Pinkus in his monograph “Ridge Functions” up to a multivariate polynomial.

It is known that for a ϱ-weighted Lq approximation of single variable functions defined on a finite or infinite interval, whose rth derivatives are in a ψ-weighted Lp space, the minimal error of approximations that use n samples of f is proportional to ‖ω1∕α‖L1α‖f(r)ψ‖Lpn−r+(1∕p−1∕q)+, where ω=ϱ∕ψ and α=r−1∕p+1∕q, provided that ‖ω1∕α‖L1<+∞. Moreover, the optimal sample points are determined by quantiles

Best approximation by the set of all n-fold linear combinations of a family of characteristic functions of measurable subsets is investigated. Such combinations generalize Heaviside-type neural networks. Existence of best approximation is studied in terms of approximative compactness, which requires convergence of distance-minimizing sequences. We show that for (Ω,μ) a measure space, in Lp(Ω,μ) with

We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of the appropriate Blaschke product at points on the unit circle.

Identifying (conditionally) strictly positive definite functions is of great importance as they allow the unique solution of certain interpolation problems. We introduce new sufficient (and some necessary) conditions for functions to be conditionally strictly positive definite on all spheres Sd−1, d>2, only employing monotonicity properties. For strictly positive definite and conditionally negative

In this article, we focus on the connections of monotonicity properties and the strict positive definiteness of functions on Rd. We collect the existing results known for the Euclidean space and present a new technique to construct positive definite functions from multiply monotone functions. Further, we collect properties of multiply monotone functions which allow us to construct new positive definite

The holomorphic extensions Fα(iz), z∈ℂd, of the Fourier transforms Fα≔f̂α of a sequence of continuous real functions fα(x), x∈Rd,α∈N0d, with exponential decay at infinity, generates a sequence of polynomials Qβ, β∈N0d, of degree |β|, that are biorthogonal to the distributional derivatives μα≔(−1)|α|fα(α), α∈N0d. The generating function is a generalized Taylor series expansion of the Fourier-Laplace

In this paper, we discuss various basic properties of moduli of smoothness of functions from Lp(Rd), 0

Four Jacobi settings are considered in the context of Hardy’s inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy’s inequality is proved for the type parameters α,β∈(−1,∞)d, whereas in the function systems for α,β∈[−1∕2,∞)d. The ranges of these parameters are the widest in which the corresponding orthonormal bases are

We study bracketing covering numbers for spaces of bounded convex functions in the Lp norms. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Bracketing number upper bounds in the supremum distance are known for bounded classes that also have a fixed Lipschitz constraint. However, in most settings of interest, the classes

We study the entropy numbers of the compact embeddings between smoothness Morrey spaces on bounded domains. Here we discover a new phenomenon when the difference of smoothness parameters in the source and target spaces is rather small compared with the influence of the fine parameters in the Morrey setting. In view of some partial forerunners this was not to be expected till now. Our argument relies

For weighted L2 spaces with doubling weights w on [−1,1] and norms ‖⋅‖2,w, the Markov factor on a polynomial P is defined by ‖P′‖2,w‖P‖2,w. We study this Markov factor on random polynomials with independent N(0,1) coefficients, and show that the upper bound of the average (expected) Markov factor is order degree to the 3∕2, as compared to the degree squared worst case upper bound.

A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equations become ordinary

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function eniλz on [−1,1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λc. Our main

We study the volume of unit balls Bp,qn of finite-dimensional Lorentz sequence spaces ℓp,qn. We give an iterative formula for vol(Bp,qn) for the weak Lebesgue spaces with q=∞ and explicit formulas for q=1 and q=∞. We derive asymptotic results for the nth root of vol(Bp,qn) and show that [vol(Bp,qn)]1∕n≍p,qn−1∕p for all 0

Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials defining the edges span a three-dimensional space of polynomials, then we compute the dimensions from the dimensions for a corresponding rectilinear