An abstract approach to Marcinkiewicz-Zygmund inequalities for approximation and quadrature in modulation spaces

Abstract

We study the approximation and quadrature error of points that satisfy Marcinkiewicz-Zygmund inequalities. First, we investigate the use of Marcinkiewicz-Zygmund inequalities in an abstract Hilbert space for an abstract approximation and quadrature rule. The setting is then specified to Sobolev spaces induced by Freud weights e − 2 σ | x | α e^{-2\sigma |x|^\alpha } with α > 1 \alpha >1 and σ > 0 \sigma >0 , and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight e − 2 π x 2 e^{-2\pi x^2} , we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis.