An abstract approach to marcinkiewicz-zygmund inequalities for approximation and quadrature in modulation spaces

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 sigma|x|alpha) with and , and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight e(-2 pi x2), we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis.