In this paper, we study a class of matrix-valued orthogonal polynomials (MVOPs) that are related to 2-periodic lozenge tilings of a hexagon. The general model depends on many parameters. In the cases of constant and 2-periodic parameter values we show that the MVOP can be expressed in terms of scalar polynomials with non-Hermitian orthogonality on a closed contour in the complex plane. The 2-periodic

This paper introduces a systematic study for analytic aspects of Fourier-Zernike series of convolutions of compactly supported functions on SE(2). Let a>0, Ba be the disk of radius a, and Da≔Ba×SO(2). Fourier-Zernike series of functions supported in domains of the form Da presented. We then apply our results to study analytic aspects of Fourier-Zernike series of convolution of functions supported in

We characterize approximation order and density order of those Parseval wavelet frames obtained from Oblique Extension Principle. These notions are closely related to approximation order and density order by a quasi-projection operator. To give our characterizations, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of a refinable function. In particular, we invoke

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions φ˜j and trigonometric polynomials φj. The class of such operators includes classical interpolation polynomials (φ˜j is the Dirac delta function), Kantorovich-type operators (φ˜j is a characteristic function), scaling expansions associated with wavelet constructions

Analysis of functions on combinatorial graphs is an emerging field attracting more and more attention. In this paper, we study the approximation of functions defined on combinatorial graphs by functions in Paley–Wiener spaces. First, we use a family of graph translation operators to define the modulus of smoothness, which has several properties similar to their counterparts in the classical approximation

The polynomials pn orthogonal on the interval [−1,1], normalized by pn(1)=1, satisfy Turán’s inequality if pn2(x)−pn−1(x)pn+1(x)≥0 for n≥1 and for all x in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Turán’s inequality. This extends essentially the results of Szwarc(1998). In particular the results can be applied to many classes of orthogonal polynomials

We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we

A distribution estimate for the representing measures of Dunkl’s intertwining operator is proved, by which some lower estimates sharper in some senses than those known for the Dunkl kernel, the associated heat kernel, and the associated Poisson kernel are obtained. We also prove some distribution estimates for the representing measures of the Dunkl-type generalized translation of radial functions.

We compute the precise value of the measure of noncompactness of Sobolev embeddings W01,p(D)↪Lp(D), p∈(1,∞), on strip-like domains D of the form Rk×∏j=1n−k(rj,qj). We show that such embeddings are always maximally noncompact, that is, their measure of noncompactness coincides with their norms. Furthermore, we show that not only the measure of noncompactness but also all strict s-numbers of the embeddings

Let C(Rn) denote the set of real valued continuous functions defined on Rn. We prove that for every n≥2 there are positive numbers λ1,…,λn and continuous functions ϕ1,…,ϕm∈C(R) with the following property: for every bounded and continuous f∈C(Rn) there is a continuous function g∈C(R) such that f(x)=∑q=1mg∑p=1nλpϕq(xp) for every x=(x1,…,xn)∈Rn. Consequently, every f∈C(Rn) can be obtained from continuous

In this paper order estimates of the Kolmogorov widths for some weighted Sobolev classes in a weighted Lebesgue space are obtained. The weighted Sobolev classes are defined by a restriction on the weighted Lp1-norm of the highest order derivative and the weighted Lp0-norm of the function.

This paper deals with orthogonal polynomials and associated connection coefficients with respect to a class of Sobolev inner products on the unit circle. Under certain conditions on the parameters in the inner product it is shown that the connection coefficients are related to a subfamily of the continuous dual Hahn polynomials. Properties regarding bounds and asymptotics are also established with

For an arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelogram containing the domain is constructed. As an application we obtain a new proof that every

Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical

In this paper we verify a sharp Bernstein type inequality for multivariate algebraic polynomials in Lp norm on general cuspidal domains which is based on a proper measure of the distance to the boundary of the domain. In particular, in case of Lipγ cuspidal graph domains the exact rate is given by n2γ−1.

Sharp bounds are given for the highest multiplicity of zeros of polynomials in terms of their norm on Jordan curves and arcs. The results extend a theorem of Erdős and Turán and solve a problem of them from 1940.

We prove, that for each r∈N, n∈N and s∈N there are a collection {yi}i=12s of points y2s0 is a constant, depending only on r. Moreover, we prove, that for each r=0,1,2 and any such collection {yi}i=12s there is a 2π - periodic function f∈C(r)(R), such that (−1)i−1f is convex on [yi,yi−1], 1≤i≤2s, and, for each sequence {Tn}n=0∞ of trigonometric polynomials Tn, satisfying (2), we have lim supn→∞nr‖f−Tn‖C(R)ω4(f(r)

We establish for dual quantization the counterpart of Kieffer’s uniqueness result for compactly supported one dimensional probability distributions having a log-concave density (also called strongly unimodal): for such distributions, Lr-optimal dual quantizers are unique at each level N, the optimal grid being the unique critical point of the quantization error. An example of non-strongly unimodal

A landmark result from rational approximation theory states that x1∕p on [0,1] can be approximated by a type-(n,n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x1∕p by composite rational functions of the form rk(x,rk−1(x,rk−2(⋯(x,r1(x,1))))). While this class

We consider a new way of factorizing the transition probability matrix of a discrete-time birth–death chain on the integers by means of an absorbing and a reflecting birth–death chain to the state 0 and viceversa. First we will consider reflecting–absorbing factorizations of birth–death chains on the integers. We give conditions on the two free parameters such that each of the factors is a stochastic

M. Derevyagin, L. Vinet and A. Zhedanov introduced in Derevyagin et al. (2012) a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real parameter λ. In Derevyagin et al. (2012) the authors prove that this map yields a natural link between the Jacobi polynomials on the unit circle and the little and big −1

We extend a higher-order sum rule proved by B. Simon to matrix valued measures on the unit circle and their matrix Verblunsky coefficients.

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by

The constrained covariance (COCO) has been proposed for measuring dependence between random vectors. Kernel cross-covariance operators on reproducing kernel Hilbert spaces, as one of kernel methods which could extract nonlinear dependence, have attracted considerable attention. This paper establishes learning rates of some estimators associated with kernel cross-covariance. For kernel cross-covariance

In this paper, we extend the sharp upper bound of Christiansen et al. (2017) and the sharp lower bound of Schiefermayr (2008) to the case of weighted Chebyshev polynomials on subsets of [−1,1] for the weight w(x)=1−x2. We then analyse the norm of Chebyshev polynomials on a circular arc, prove monotonicity of the corresponding Widom factors, find exact values of their supremum and infimum, and obtain

A class of Schrödinger-type second-order linear differential equations with a large parameter u is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of u. These formal series solutions are called the WKB solutions. We show that under mild conditions on the potential function of the equation, the WKB solutions are Borel summable

We show that, for any integer n≥2, the space C(Q) (where Q is a Hausdorff compact set, card Q>n) contains an n-dimensional subspace such that any translation thereof by a vector p, ‖p‖<1, intersects the unit ball B of C(Q) in a nonsmooth set. In L1[0,1], we show that if ℓ is an arbitrary finite-dimensional subspace in L1[0,1], dimℓ⩾1, then there exists a dense set in the unit ball B⊂L1[0,1] set of

It is known by a formula of Hasse–Sondow that the Riemann zeta function is given, for any s=σ+it∈ℂ, by ∑n=0∞A˜(n,s) where A˜(n,s)≔12n+1(1−21−s)∑k=0nnk(−1)k(k+1)s. For any N∈N, we prove the following approximate functional equation for the Hasse–Sondow presentation: ζ(s)=∑n≤|t|πNA˜(n,s)+χ(s)1−2s−1∑k≤N(2k−1)s−1+Oe−ω(N)t, where ω(N)≈e−4Ne. The proof is based on a study, via saddle point techniques, of

We prove that the sharp constant in the univariate Bernstein–Nikolskii inequality for entire functions of exponential type is the limit of the sharp constant in the V. A. Markov type inequality with an exponential weight for coefficients of an algebraic polynomials of degree n as n→∞.

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)

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 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 Muckenhoupt weights.

We first give a method to get multidimensional Leja sequences by considering intertwining sequences from one-dimensional ones. An application is the existence of explicit Leja sequences for the closed unit polydisc. Next, we deal with some applications in bidimensional Lagrange interpolation with intertwining Leja sequences. These results also require an explicit formula for the associated fundamental

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

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

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

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