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

Ehler, Martin; Gröchenig, Karlheinz

doi:10.1090/mcom/3930

An abstract approach to Marcinkiewicz-Zygmund inequalities for approximation and quadrature in modulation spaces
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by Martin Ehler and Karlheinz Gröchenig;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/3930

Published electronically: December 20, 2023

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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 $e^{-2\sigma |x|^\alpha }$ with $\alpha >1$ and $\sigma >0$, and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight $e^{-2\pi x^2}$, we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis.

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