Recovering functions from the modulation spaces $\mathscr{F}W$

TL;DR

This paper demonstrates that functions in a specific modulation space can be recovered similarly to band-limited functions using cardinal interpolators, with convergence in norm and pointwise.

Contribution

It establishes that modulation space functions can be reconstructed via cardinal interpolation, extending classical recovery results to this broader function space.

Findings

Functions in the modulation space $\

$f$ can be recovered from cardinal interpolators in norm and pointwise.

Recovery methods for band-limited functions extend to functions in the modulation space $\

Abstract

In this short note we show that functions in the modulation space $\mathscr{F}W=\{ f: \sum_{j\in\mathbb{Z}^n}\| \hat{f}(\cdot+2\pi j)\|_{L_\infty([-\pi,\pi]^n)}<\infty \}$ enjoy similar recovery properties as band-limited functions. If $\{\phi_\alpha\}$ is a regular family of cardinal interpolators, then one can build an approximand of $f$ using the fundamental functions corresponding to $\phi_\alpha$. Then taking the appropriate limit, one recovers $f$ both in norm and pointwise.