Gibbs phenomenon and its removal for a class of orthogonal expansions

We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.     

Orthogonal Structure on a Wedge and on the Boundary of a Square

Orthogonal polynomials with respect to a weight function defined on a wedge in the plane are studied. A basis of orthogonal polynomials is explicitly constructed for two large class of weight functions and the convergence of Fourier orthogonal expansions is studied. These are used to establish analogous results for orthogonal polynomials on the boundary of the square. As an application, we study the statistics of the associated determinantal point process and use the basis to calculate Stieltjes transforms.

     

A summability for Meyer wavelets

The Gibbs’ phenomenon in the classical Fourier series is well-known. It is closely related with the kernel of the partial sum of the series. In fact, the Dirichlet kernel of the Fourier series is not positive. The poisson kernel of Cesaro summability is positive. As the consequence of the positiveness, the partial sum of Cesaro summability does not exhibit the Gibbs’ phenomenon. Most kernels associated with wavelet expansions are not positive. So wavelet series is not free from the Gibbs’ phenomenon. Because of the excessive oscillation of wavelets, we can not follow the techniques of the Fourier series to get rid of the unwanted quirk. Here we make a positive kernel for Meyer wavelets and as the result the associated summability method does not exhibit Gibbs’ phenomenon for the corresponding series.     

Spectral convergence for orthogonal polynomials on triangles

We study the behavior of orthogonal polynomials on triangles and their coefficients in the context of spectral approximations of partial differential equations. In these spectral approximations one studies series expansions $u=\sum _{k=0}^{\infty } \hat{u}_k \phi _k$ where the $\phi _k$ are orthogonal polynomials. Our results show that for any function $u\in C^{\infty }$ the series expansion converges faster than with polynomial order.     

Gibbs' phenomenon for sampling series and what to do about it

Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.     

THE GIBBS PHENOMENON,THE PINSKY PHENOMENON,AND VARIANTS FOR EIGENFUNCTION EXPANSIONS

ABSTRACT

We examine analogues of the Gibbs phenomenon for eigenfunction expansions of functions with singularities across a smooth surface, though of a more general nature than a simple jump. The Gibbs phenomena that arise still have a universal form, but a more general class of “fractional sine integrals” arises, and we study these functions. We also make a uniform analysis of eigenfunction expansions in the presence of the Pinsky phenomenon, and see an analogue of the Gibbs phenomenon there. These analyses are done on three classes of manifolds: strongly scattering manifolds, including Euclidean space; compact manifolds without strongly focusing geodesic flows, including flat tori and quotients of hyperbolic space, and compact manifolds with periodic geodesic flow; including spheres and Zoll surfaces. Finally, we uncover a new divergence phenomenon for eigenfunction expansions of characteristic functions of balls, for a perturbation of the Laplace operator on a sphere of dimension ≥5.     

Expansions of one density via polynomials orthogonal with respect to the other

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam-Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson-Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets.     

On Legendre functions of imaginary degree and associated integral transforms

New integral representations, asymptotic formulas, and series expansions in powers of tanh(t/2) are obtained for the imaginary and real parts of the Legendre function P_iξ(cosht). Coefficients of these series expansions are orthogonal polynomials in the real variable ξ. A number of relations for these orthogonal polynomials are obtained on the basis of the generating function. Several inversion theorems are proven for the integral transforms involving the Legendre function of imaginary degree. In many cases it is preferable to employ these transforms, than Mehler-Fok transforms, since conditions placed on functions are less restrictive.     

Sub-exponentially localized kernels and frames induced by orthogonal expansions

The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions.     

Hybrid sampling series associated with orthogonal wavelets and Gibbs phenomenon

When a sampling theorem holds in wavelet subspaces, sampling expansions can be a good approximation to projection expansions. Even when the sampling theorem does not hold, the scaling function series with the usual coefficients replaced by sampled function values may also be a good approximation to the projection. We refer to such series as hybrid sampling series. For this series, we shall investigate the local convergence and analyze Gibbs phenomenon.     

A General Approach to Gibbs Phenomenon

Gibbs phenomenon occurs for most approximations based on standard orthogonal expansions, as well as for those based on integral operators. It also occurs in interpolations and other types of approximations. We consider a general approach to approximation based on delta sequences in an attempt to better understand the concept.     

Calculation of expansion coefficients of series in Chebyshev polynomials for a solution to a Cauchy problem

Two approaches are proposed to determine an initial approximation for the coefficients of an expansion of the solution to a Cauchy problem for ordinary differential equations in the form of series in shifted Chebyshev polynomials of the first kind. This approximation is used in an analytical method to solve ordinary differential equations using orthogonal expansions.     

Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials

It has been shown in Ferreira et al. [Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials, Adv. in Appl. Math. 31(1) (2003) 61–85], López and Temme [Approximations of orthogonal polynomials in terms of Hermite polynomials, Methods Appl. Anal. 6 (1999) 131–146; The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis, J. Comput. Appl. Math. 133 (2001) 623–633] that the three lower levels of the Askey table of hypergeometric orthogonal polynomials are connected by means of asymptotic relations. In Ferreira et al. [Limit relations between the Hahn polynomials and the Hermite, Laguerre and Charlier polynomials, submitted for publication] we have established new asymptotic connections between the fourth level and the two lower levels. In this paper, we continue with that program and obtain asymptotic expansions between the fourth level and the third level: we derive 16 asymptotic expansions of the Hahn, dual Hahn, continuous Hahn and continuous dual Hahn polynomials in terms of Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials. From these expansions, we also derive three new limits between those polynomials. Some numerical experiments show the accuracy of the approximations and, in particular, the accuracy in the approximation of the zeros of those polynomials.     

A formula for polynomials with Hermitian matrix argument

We construct and study orthogonal bases of generalized polynomials on the space of Hermitian matrices. They are obtained by the Gram-Schmidt orthogonalization process from the Schur polynomials. A Berezin-Karpelevich type formula is given for these multivariate polynomials. The normalization of the orthogonal polynomials of Hermitian matrix argument and expansions in such polynomials are investigated.     

Flat Power Series over a Finite Field

We define and describe a class of algebraic continued fractions for power series over a finite field. These continued fraction expansions, for which all the partial quotients are polynomials of degree one, have a regular pattern induced by the Frobenius homomorphism.This is an extension, in the case of positive characteristic, of purely periodic expansions corresponding to quadratic power series.     

Asymptotics for orthogonal polynomials defined by a recurrence relation

Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.     

Asymptotics of orthogonal polynomials with asymptotic Freud-like weights

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.     

Extraction of orthogonal polynomials from generating function for reciprocal of odd numbers

In the present paper the orthogonality relations, exhibited by both numerator and denominator polynomials of both even and odd order convergents of a regular C-fraction of a power series with coefficients as reciprocal of odd numbers are described. The two sequences of convergents are nothing but diagonal and upper diagonal Pade approximants for the power series. The two orthogonal polynomials extracted from denominators are shown to be classical orthogonal polynomials and two orthogonal polynomials extracted from numerators are shown to be non-classical orthogonal polynomials..     

Marcinkiewicz–Zygmund type results in multivariate domains

We investigate Marcinkiewicz–Zygmund type inequalities for multivariate polynomials on various compact domains in \({\mathbb{R}^d}\). These inequalities provide a basic tool for the discretization of the L ^p norm and are widely used in the study of the convergence properties of Fourier series, interpolation processes and orthogonal expansions. Recently Marcinkiewicz–Zygmund type inequalities were verified for univariate polynomials for the general class of doubling weights, and for multivariate polynomials on the ball and sphere with doubling weights. The main goal of the present paper is to extend these considerations to more general multidimensional domains, which in particular include polytopes, cones, spherical sectors, toruses, etc. Our approach will rely on application of various polynomial inequalities, such as Bernstein–Markov, Schur and Videnskii type estimates, and also using symmetry and rotation in order to generate results on new domains.     

L2-Theory of Riesz Transforms for Orthogonal Expansions

We propose a unified approach to the theory of Riesz transforms and conjugacy in the setting of multi-dimensional orthogonal expansions. The scheme is supported by numerous examples concerning, in particular, the classical orthogonal expansions in Hermite, Laguerre, and Jacobi polynomials. A general case of expansions associated to a regular or singular Sturm-Liouville problem is also discussed.