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

We consider optimization problems associated with a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function \(f: \mathbb {R}\rightarrow \mathbb {R}\) of the form $$\begin{aligned} x(k+1) = f(x(k)) + u(k), \end{aligned}$$ where $$\begin{aligned}

We prove Menshov type “correction” theorems for sequences of compact operators, recovering several results of Fourier series in trigonometric and Walsh systems. The paper clarifies the main ingredient that is important in the study of such “correction” theorems. That is the weak-\(L^1\) estimate for the maximal Fourier sums of indicator functions of some specific sets.

Asymptotic expansions are obtained for contour integrals of the form $$\begin{aligned} \int _a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right) q(t)\mathrm{d}t, \end{aligned}$$ in which z is a large real or complex parameter; p(t), q(t), and r(t) are analytic functions of t; and the positive constants \(\mu \) and \(\nu \) are related to the local behavior of the functions p(t) and r(t) near the

We investigate type I multiple orthogonal polynomials on r intervals that have a common point at the origin and endpoints at the r roots of unity \(\omega ^j\), \(j=0,1,\ldots ,r-1\), with \(\omega = \exp (2\pi i/r)\). We use the weight function \(|x|^\beta (1-x^r)^\alpha \), with \(\alpha ,\beta >-1\), for the multiple orthogonality relations. We give explicit formulas for the type I multiple orthogonal

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan R-function) R(a), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of R(a), \(\sin (\pi a)\) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented

Upper bounds for the \(L_p\)-discrepancies of point distributions in compact metric measure spaces are proved for all exponents \(0

We prove the existence of modified wave operators for one-dimensional Dirac operators whose spectral measures belong to the Szegő class on the real line.

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

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

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.

The Painlevé-III equation with parameters \(\Theta _0=n+m\) and \(\Theta _\infty =m-n+1\) has a unique rational solution \(u(x)=u_n(x;m)\) with \(u_n(\infty ;m)=1\) whenever \(n\in \mathbb {Z}\). Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018), we study the asymptotic behavior of \(u_n(x;m)\) in the limit \(n\rightarrow +\infty \) with \(m\in \mathbb

We prove in this paper one weight norm inequalities for some positive Bergman-type operators.

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

We present a rational extension of the Newton diagram for the positivity of \({}_1F_2\) generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots \(\beta (\alpha )\) of $$\begin{aligned} \int _0^{j_{\alpha , 2}} t^{-\beta } J_\alpha (t) \hbox {d}t = 0\qquad (-\,1<\alpha \le 1/2), \end{aligned}$$ where \(j_{\alpha , 2}\) denotes the second positive

The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in \({{\mathbb {R}}}^n\), and in particular on the interval, generated by classical differential operators whose eigenfunctions are algebraic polynomials. To this end we develop a general method that employs the natural relation of such operators with weighted Laplace operators on suitable

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

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

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

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

Let Ω be a bounded closed convex set in ℝ d with non-empty interior, and let 𝒞 r (Ω) be the class of convex functions on Ω with Lr -norm bounded by 1. We obtain sharp estimates of the ε-entropy of 𝒞 r (Ω) under Lp (Ω) metrics, 1 ≤ p < r ≤ ∞. In particular, the results imply that the universal lower bound ε-d/2 is also an upper bound for all d-polytopes, and the universal upper bound of [Formula:

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

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.

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

This paper aims at finding conditions on a Hamburger or Stieltjes moment sequence, under which the change of at most a finite number of its entries produces another sequence of the same type. It turns out that a moment sequence allows all small enough variations of this kind precisely when it is indeterminate. We also show that a determinate moment sequence has the finite index of determinacy if and

Recently we introduced mixed-state localization operators associated with a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution that is based on the Cohen class distribution associated with the density operator and the eigenvectors of the mixed-state localization

Our aim in this paper is to attain a sharp asymptotic estimate for the greedy constant \(C_{g}[\mathcal {H}^{(p)},L_p]\) of the (normalized) Haar system \(\mathcal {H}^{(p)}\) in \(L_{p}[0,1]\) for \(1

Morrey (function) spaces and, in particular, smoothness spaces of Besov–Morrey or Triebel–Lizorkin–Morrey type have enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces \(m_{u,p}=m_{u,p}(\mathbb {Z}^d)\), \(0