On Spectral Approximations With Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

TL;DR

This paper investigates spectral approximation methods using nonstandard weight functions for multidimensional orthogonal polynomials, establishing convergence rates and demonstrating applications to generalized chaos expansions and efficient integration of singular functions.

Contribution

It introduces a convergence analysis and a spectrally convergent multidimensional integration method for orthogonal polynomials with nonstandard weights, applicable to generalized chaos expansions.

Findings

Convergence rate determined via a comparison lemma.

Spectrally convergent multidimensional integration method developed.

Effective integration of singular functions demonstrated.

Abstract

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm-Liouville problem, as is the case for classical polynomials. Therefore, standard results regarding convergence cannot be applied. Furthermore, since in general, the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a tensor product of one-dimensional orthogonal polynomials, as well. In this work, we determine the convergence rate using a comparison Lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We show that the use of nonstandard weight functions can allow for efficient integration of singular functions. We also apply this method to Generalized Polynomial Chaos Expansions in the case of dependent random variables.