We consider discrete Dirac systems as an approach to the study of the corresponding block Toeplitz matrices, which in many ways completes the famous approach via Szegő recurrences and matrix orthogonal polynomials. We prove an analog of the Christoffel–Darboux formula and derive the asymptotic relations for the analog of reproducing kernel (using Weyl–Titchmarsh functions of discrete Dirac systems)

We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz s-energy on the sphere \(\mathbb S^d.\) Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where estimates for the spherical cap discrepancy of the logarithmic energy minimizers in \(\mathbb S^2\) were obtained. Our result improves previously known

In this paper, we continue the study of Lebesgue-type inequalities for greedy algorithms. We introduce the notion of strong partially greedy Markushevich bases and study the Lebesgue-type parameters associated with them. We prove that this property is equivalent to that of being conservative and quasi-greedy, extending a similar result given in Dilworth et al. (Constr Approx 19:575–597, 2003) for Schauder

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms \(L_q\), \(1\le q<\infty \), which is an extension of known results for \(q=1\). To discretize the integral norms

We provide a new method to approximate a (possibly discontinuous) function using Christoffel–Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to nonlinear

The aim of this paper is to prove that Markov’s theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle.

Gaussians are a family of Mercer kernels which are widely used in machine learning and statistics. The variance of a Gaussian kernel reflexes the specific structure of the reproducing kernel Hilbert spaces (RKHS) induced by the Gaussian or other important features of learning problems such as the frequency of function components. As the variance of the Gaussian decreases, the learning performance and

As for the \({}_2F_3\) hypergeometric function of the form $$\begin{aligned} {}_2F_3\left[ \begin{array}{c} a_1, a_2\\ b_1, b_2, b_3\end{array}\biggr | -x^2\right] \qquad (x>0), \end{aligned}$$ where all of parameters are assumed to be positive, we give sufficient conditions on \((b_1, b_2, b_3)\) for its positivity in terms of Newton polyhedra with vertices consisting of permutations of \(\,(a_2,

We establish a Wiman–Valiron theory of a polynomial series based on the Askey–Wilson operator \({\mathcal {D}}_q\), where \(q\in (0,1)\). For an entire function f of log-order smaller than 2, this theory includes (i) an estimate which shows that f behaves locally like a polynomial consisting of the terms near the maximal term of its Askey–Wilson series expansion, and (ii) an estimate of \({\mathcal

Sobolev wavefront sets and 2-microlocal spaces play a key role in describing and analyzing the singularities of distributions in microlocal analysis and solutions of partial differential equations. Employing the continuous shearlet transform to Sobolev spaces, in this paper we characterize the microlocal Sobolev wavefront sets, the 2-microlocal spaces, and local Hölder spaces of distributions/functions

Let \(W_0(\mathbb {R})\) be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proved in the paper that, in particular, a trigonometric series \(\sum \nolimits _{k=-\infty }^\infty c_k e^{ikt}\) is the Fourier series of an integrable function if and only if there exists a \(\phi \in W_0(\mathbb {R})\) such that \(\phi (k)=c_k\), \(k\in

We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calderón and Torchinsky, and Grafakos, He, Honzík, and Nguyen, and an improvement of the result of Triebel. For \(0\frac{d}{s-(d/\min {(1,p,q)}-d)}\), then $$\begin{aligned} \big \Vert \big \{\big ( m_k \widehat{f_k}\big )^{\vee }\big \}_{k\in {\mathbb {Z}}}\big \Vert _{L^p(\ell ^q)}\lesssim

We prove that for entire functions f of finite order, there is a sequence of integers \(\mathcal {S}\) such that as \(n\rightarrow \infty \) through S, $$\begin{aligned} \min \left\{ \left| f-\left[ n/n\right] \right| \left( z\right) ,\left| f-\left[ n-1/n-1\right] \right| \left( z\right) \right\} ^{1/n}\rightarrow 0 \end{aligned}$$ uniformly for z in compact subsets of the plane. More generally this

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining

We show that the problem of finding the measure supported on a compact set \(K\subset \mathbb {C}\) such that the variance of the least squares predictor by polynomials of degree at most n at a point \(z_0\in \mathbb {C}^d\backslash K\) is a minimum is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K, with extremal growth at \(z_0.\) We use this to find the

Let \(({\mathcal {X}},\rho , \mu )\) be a space of homogeneous type. Suppose that \(p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]\) are such that both \(1/p(\cdot )\) and \(1/q(\cdot )\) satisfy the globally log-Hölder continuous condition, and \(s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R\) is a bounded function satisfying the locally log-Hölder continuous condition. In this article

For Jacobi parameters belonging to one of three classes: asymptotically periodic, periodically modulated, and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel–Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports

With the sphere \(\mathbb {S}^2 \subset \mathbb {R}^3\) as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials \(C_n^{(1+\alpha )}(z)\) for \(\alpha >-1\) containing the Legendre polynomials \(P_n(z)\) and the subset \(P_n^{(\alpha +\frac{1}{2},\pm \frac{1}{2})}(z)\)

We establish a sharp upper bound for the absolute value of the derivative of the finite Blaschke product, provided that the critical values of this product lie in a given disk.

We apply the method of moments to prove a recent conjecture of Haikin, Zamir and Gavish concerning the distribution of the singular values of random subensembles of Paley equiangular tight frames. Our analysis applies more generally to real equiangular tight frames of redundancy 2, and we suspect similar ideas will eventually produce more general results for arbitrary choices of redundancy.

We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups.

We study the embedding \(\mathrm {id}: \ell _p^b(\ell _q^d) \rightarrow \ell _r^b(\ell _u^d)\) and prove matching bounds for the entropy numbers \(e_k(\mathrm {id})\) provided that \(0

In this paper we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J. Brauchart, P. Grabner, W. Kusner and J. Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls, or balls of threshold order on the flat tori are uniformly distributed. Moreover, it is also shown that QMC-designs sequences for Sobolev classes, probabilistic

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces \({{\mathcal {P}}}_\Lambda \) described by lower sets \(\Lambda \). Given

We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the q-lattice \(\{q^k, k=0,1,2,3,\ldots \}\), where \(0< q < 1\). The asymptotic distribution of the zeros is given by the radial part of the equilibrium measure of an extremal problem in logarithmic potential theory for circular symmetric measures with

Let \(\mathcal{P}_n^c\) denote the set of all algebraic polynomials of degree at most n with complex coefficients. Let $$\begin{aligned} D^+ := \{z \in \mathbb {C}: |z| \le 1, \text { Im}(z) \ge 0\}\,. \end{aligned}$$ For integers \(0 \le k \le n\) let \(\mathcal{F}_{n,k}^c\) be the set of all polynomials \(P \in \mathcal{P}_n^c\) having at least \(n-k\) zeros in \(D^+\). Let $$\begin{aligned} \Vert

In this paper, we provide an efficient method for computing the Taylor coefficients of \(1-p_n f\), where \(p_n\) denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space \(H^2_\omega \) of analytic functions over the unit disc \(\mathbb {D}\), and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces \(H^2_\omega \), the sequence

In a previous paper by the author, a family of iterations for computing the matrix square root was constructed by exploiting a recursion obeyed by Zolotarev’s rational minimax approximants of the function \(z^{1/2}\). The present paper generalizes this construction by deriving rational minimax iterations for the matrix pth root, where \(p \ge 2\) is an integer. The analysis of these iterations is considerably

We establish analogues of the Hermite-Poulain theorem for linear finite difference operators with constant coefficients defined on sets of polynomials with roots on a straight line, in a strip, or in a half-plane. We also consider the central finite difference operator of the form $$\begin{aligned} \Delta _{\theta , h}(f)(z)=e^{i\theta }f(z+ih)-e^{-i\theta }f(z-ih), \quad \theta \in [0,\pi ),\ \ h\in

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group \(\mathrm {SO}(3)\), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on \(\mathrm {SO}(3)\). The variance of rotation matrices sampled by a certain

We recall a uniqueness theorem of E. B. Vul pertaining to a version of the cosine transform originating in spectral theory. Then we point out an application to the Bernstein approximation problem with non-symmetric weights: a theorem of Volberg is proved by elementary means.

This paper considers the problem of restricting the short-time Fourier transform to sets of nonzero measure in the plane. Thereby, we study under which conditions one has a sampling set and provide estimates of the corresponding sampling bound. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class

In this paper, we present a generalization of Lehmann’s approach for solving approximation problems on hypersurfaces to situations with arbitrary codimension. We show that as in the case of hypersurfaces, the method is able to transfer approximation orders from the ambient space to the submanifold. In particular, the resulting approximant is \({\mathrm {C}}^{m-2}\) and the error decays at an optimal

The generalized Prony method is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator A. However, this procedure requires the evaluation of higher powers of the linear operator A that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method

We study scaling limits of deterministic Jacobi matrices, centered around a fixed point \(x_0\), and their connection to the scaling limits of the Christoffel–Darboux kernel at that point. We show that in the case when the orthogonal polynomials are bounded at \(x_0\), a subsequential limit always exists and can be expressed as a canonical system. We further show that under weak conditions on the associated

Rhodonea curves are classical planar curves in the unit disk with the characteristic shape of a rose. In this work, we use these rose curves as sampling trajectories to create novel nodes for spectral interpolation on the disk. By generating the interpolation spaces with a parity-modified Chebyshev–Fourier basis, we will prove the unisolvence of the interpolation on the rhodonea nodes. Properties such

Let \(1\leqslant p<\infty \), \(0

This paper is a condensation of my arXiv monograph entitled Schwartz “The phase transition in 5 point energy minimization”, 2016. arXiv:1610.03303, which contains a complete proof that there is a constant such that the triangular bi-pyramid is the minimizer, amongst all 5 point configurations on the sphere, with respect to the power law potential \(R_s(r)=\mathrm{sign}(s)/r^s\), if and only if . In

Motivated by the problem of determining the values of \(\alpha >0\) for which \(f_\alpha (x)=\mathrm{e}^\alpha - (1+1/x)^{\alpha x},\ x>0\), is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family \(\varphi _\alpha \), \(\alpha >0\), of entire functions such that \(f_\alpha (x) =\int _0^\infty \mathrm{e}^{-sx}\varphi _\alpha (s)\,\mathrm{d}s, \ x>0.\)

We continue the study of the variable exponent Morreyfied Triebel–Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions

We correct the expression for the worst-case error derived in [Kuo, Wasilkowski, Woźniakowski, Construct. Approx. 30 (2009), 475–493] and explain that the main theorem of the paper holds with enlarged constants.

We study the minimality properties of a new type of “soft” theta functions. For a lattice \(L\subset {\mathbb {R}}^d\), an L-periodic distribution of mass \(\mu _L\), and another mass \(\nu _z\) centered at \(z\in {\mathbb {R}}^d\), we define, for all scaling parameters \(\alpha >0\), the translated lattice theta function \(\theta _{\mu _L+\nu _z}(\alpha )\) as the Gaussian interaction energy between

Consider the Hermite operator \(H=-\Delta +|x|^2\) on the Euclidean space \({\mathbb {R}}^n\). The aim of this article is to prove the boundedness of higher-order Riesz trnsforms on appropriate Besov and Triebel–Lizorkin spaces. As an application, we prove certain regularity estimates of second-order elliptic equations in divergence form with the oscillator perturbations.

The Littlewood–Paley theory of homogeneous product Besov and Triebel–Lizorkin spaces is developed in the spirit of the \(\varphi \)-transform of Frazier and Jawerth. This includes the frame characterization of the product Besov and Triebel–Lizorkin spaces and the development of almost diagonal operators on these spaces. The almost diagonal operators are used to obtain product wavelet decomposition

It is shown that if$$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$where R(z, f) is rational in f with meromorphic coefficients and \(\deg _f(R(z,f))=n\), has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid

In order to avoid the curse of dimensionality, frequently encountered in big data analysis, there has been vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower-dimensional manifold; thus the high dimensionality problem can be overcome

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have

The Dunkl operators associated with a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in \({\mathbb {R}}^2\). The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of

In this paper, we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples.

We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov–Bernstein–Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in \({\mathbb {R}}^m\) and the corresponding constants for entire functions of spherical type on \({\mathbb

Let K be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for K are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized

A domain \(\Theta _n\) is called a Rolle’s domain if every complex polynomial p of degree n, satisfying \(p(i)=p(-i)\), has at least one critical point in it. In this paper, we find the smallest possible Rolle’s domain made up of two closed disks that are symmetric with respect to the real and the imaginary axes. This is a strengthening of the main result in Sendov and Sendov (Proc Am Math Soc 146(8):3367–3380

In analog-to-digital conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the \(\Sigma \Delta \) quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results are proved for finite unitarily generated frames. We introduce

This article describes an elementary construction of a dual basis for nonuniform B-splines that is local, \(L^\infty \)-stable, and reproduces polynomials of any prescribed degree. This allows one to define local projection operators with near-optimal approximation properties in any \(L^q\), \(1 \le q \le \infty \), and high order moment preserving properties. As the dual basis functions share the

The purpose of this paper is to provide necessary and sufficient conditions on a continuous and matrix valued radial kernel on a Euclidean space in order that it be conditionally positive definite of a fixed order. Except for the one dimensional Euclidean space, the strict conditional positive definiteness of the kernel is fully characterized.

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We

In previous work, a higher rank generalization R(n) of the Racah algebra was defined abstractly. The special case of rank one encodes the bispectrality of the univariate Racah polynomials and is known to admit an explicit realization in terms of the operators associated with these polynomials. Starting from the Dunkl model for which we have an action by R(n) on the Dunkl-harmonics, we show that connection

We discuss the classical problem of how to pick N weighted points on a d-dimensional manifold so as to obtain a reasonable quadrature rule$$\begin{aligned} \frac{1}{|M|}\int _{M}{f(x) \mathrm{d}x} \simeq \sum _{n=1}^{N}{a_i f(x_i)}. \end{aligned}$$This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional$$\begin{aligned}