Gauss Quadrature for Freud Weights, Modulation Spaces, and Marcinkiewicz-Zygmund Inequalities

TL;DR

This paper investigates Gauss quadrature for Freud weights, providing error estimates for Sobolev and modulation spaces, and extends results to more general nodes based on Marcinkiewicz-Zygmund inequalities, highlighting stability aspects.

Contribution

It introduces new error bounds for Gauss quadrature with Freud weights and extends the analysis to general nodes using Marcinkiewicz-Zygmund inequalities, linking quadrature stability with function space properties.

Findings

Error estimates for Freud weight quadrature in Sobolev spaces

Identification of modulation spaces with Gaussian weights in analysis

Extension to general nodes via Marcinkiewicz-Zygmund inequalities

Abstract

We study Gauss quadrature for Freud weights and derive worst case error estimates for functions in a family of associated Sobolev spaces. For the Gaussian weight $e^{-\pi x^2}$ these spaces coincide with a class of modulation spaces which are well-known in (time-frequency) analysis and also appear under the name of Hermite spaces. Extensions are given to more general sets of nodes that are derived from Marcinkiewicz-Zygmund inequalities. This generalization can be interpreted as a stability result for Gauss quadrature.