In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence \((P_n(x))_{n\in \mathbb {N}_0}\) satisfies nonnegative linearization of products, i.e., the product of any two \(P_m(x),P_n(x)\) is a conical combination of the polynomials \(P_{|m-n|}(x),\ldots ,P_{m+n}(x)\). Since the

We address the structure identification and the uniform approximation of two fully nonlinear layer neural networks of the type \(f(x)=1^T h(B^T g(A^T x))\) on \(\mathbb R^d\), where \(g=(g_1,\dots , g_{m_0})\), \(h=(h_1,\dots , h_{m_1})\), \(A=(a_1|\dots |a_{m_0}) \in \mathbb R^{d \times m_0}\) and \(B=(b_1|\dots |b_{m_1}) \in \mathbb R^{m_0 \times m_1}\), from a small number of query samples. The

The goal of this paper is threefold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy—the smooth discrepancy. Third, we consider numerical integration in classes, for which we

We show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. Given a training set \(s_1, \ldots , s_n \in {\mathbb {R}}^p\) with corresponding responses \(t_1,\ldots ,t_n \in {\mathbb {R}}^q\), fitting a k-layer neural network \(\nu _\theta : {\mathbb {R}}^p \rightarrow {\mathbb {R}}^q\) involves estimation of the weights \(\theta \in {\mathbb {R}}^m\)

Existing depth separation results for constant-depth networks essentially show that certain radial functions in \(\mathbb {R}^d\), which can be easily approximated with depth 3 networks, cannot be approximated by depth 2 networks, even up to constant accuracy, unless their size is exponential in d. However, the functions used to demonstrate this are rapidly oscillating, with a Lipschitz parameter scaling

We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual

This paper addresses the following question of neural network identifiability: Does the input–output map realized by a feed-forward neural network with respect to a given nonlinearity uniquely specify the network architecture, weights, and biases? The existing literature on the subject (Sussman in Neural Netw 5(4):589–593, 1992; Albertini et al. in Artificial neural networks for speech and vision,

We analyze approximation rates by deep ReLU networks of a class of multivariate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReLU architectures capable of efficiently approximating tensor products. Combining this with results concerning the approximation of well-behaved (i.e., fulfilling some smoothness properties) univariate functions, this provides

We study the expressivity of deep neural networks. Measuring a network’s complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. Using results from classical approximation theory, we show that this class can be

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are both invariant/equivariant and provably complete in the sense of their ability to approximate any continuous invariant/equivariant map. Our contribution is three-fold

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g., approximation by polynomials or

The power series expansions of the hypergeometric functions \({}_{p+1}F_p(a,b_1,\ldots ,b_p;c_1,\ldots ,c_p;z)\) converge either inside the unit disk \(\vert z\vert <1\) or outside this disk \(\vert z\vert >1\). Nørlund’s expansion in powers of \(z/(z-1)\) converges in the half-plane \(\mathfrak {R}(z)<1/2\). For arbitrary \(z_0 \in \mathbb {C}\), Bühring’s expansion in inverse powers of \(z-z_0\)

For a parameter dimension \(d\in {\mathbb {N}}\), we consider the approximation of many-parametric maps \(u: [-\,1,1]^d\rightarrow {\mathbb R}\) by deep ReLU neural networks. The input dimension d may possibly be large, and we assume quantitative control of the domain of holomorphy of u: i.e., u admits a holomorphic extension to a Bernstein polyellipse \({{\mathcal {E}}}_{\rho _1}\times \cdots \times

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 subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to 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 this external point. We use this to find the polynomials

Let $$({\mathcal {X}},\rho , \mu )$$ ( X , ρ , μ ) be a space of homogeneous type. Suppose that $$p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]$$ p ( · ) , q ( · ) : X → ( 0 , ∞ ] are such that both $$1/p(\cdot )$$ 1 / p ( · ) and $$1/q(\cdot )$$ 1 / q ( · ) satisfy the globally log-Hölder continuous condition, and $$s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R$$ s ( · ) : X → R

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 the

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 $P_n^{(\alpha+\frac12,\pm\frac12)}(z)$ of the Jacobi

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 (2017) 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 $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ 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 $\cal{P}_{\Lambda}$ described by lower sets $\Lambda$. Given a budget $n$

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 ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For integers $0 \leq k \leq n$ 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 $$\|f\|_A :=

We compute the Taylor coefficients of $p_n f-1$, 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 an application, we show that the sequence $\{p_nf-1\}_{n\in \mathbb{N}}$ is uniformly bounded on the closed unit disc and,

In [E. S. Gawlik, Zolotarev iterations for the matrix square root, arXiv preprint 1804.11000, (2018)], 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 $p^{th}$ root

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 $$\mathrm {SO}(3)$$ 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 $$\mathrm {SO}(3)$$ SO ( 3 ) . The variance of rotation matrices

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 introduced by Peter & Plonka (2013) 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

We study scaling limits of deterministic Jacobi matrices, centered around a fixed point $$x_0$$ 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 $$x_0$$ x 0 , a subsequential limit always exists and can be expressed as a canonical system. We further show that under weak conditions

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 $$R_s(r)=\mathrm{sign}(s)/r^s$$ R s ( r ) = sign (

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 $$1\leqslant p<\infty $$ 1 ⩽ p < ∞ , $$0

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

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 $$L\subset {\mathbb {R}}^d$$ L ⊂ R d , an L -periodic distribution of mass $$\mu _L$$ μ L , and another mass $$\nu _z$$ ν z centered at $$z\in {\mathbb {R}}^d$$ z ∈ R d , we define, for all scaling parameters $$\alpha >0$$ α > 0 , the translated lattice theta function $$\theta _{\mu _L+\nu _z}(\alpha )$$ θ μ L

Consider the Hermite operator $$H=-\Delta +|x|^2$$ H = - Δ + | x | 2 on the Euclidean space $${\mathbb {R}}^n$$ 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 $$\varphi $$ φ -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