Recovering functions from the modulation spaces mathscr{F}W

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.