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Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} |\cdot|^2} \in L^p(\mathbb{C}^n, dv) $ where $dv$ is ordinary Lebesgue volume measure and where $F_\alpha ^p (\mathbb{C}^n)$ is given it's natural Banach space norm.

For $\epsilon > 0$ small, treat $\epsilon \mathbb{Z}^{2n}$ canonically as a lattice in $\mathbb{C}^n$. It is well known that $\epsilon \mathbb{Z}^{2n}$ is a "sampling set" for $F_\alpha ^p (\mathbb{C}^n)$, so that there exists constants $C_1, C_2 > 0$ independent of $f \in F_\alpha ^p (\mathbb{C}^n)$ where

\begin{align} C_1 \sum_{\sigma \in \epsilon \mathbb{Z}^{2n} } |f(\sigma)|^p e^{- \frac{\alpha p |\sigma|^2}{2} } \leq \|f\|_ {F_\alpha ^p (\mathbb{C}^n) } ^p \leq C_2 \sum_{\sigma \in \epsilon \mathbb{Z}^{2n} } |f(\sigma)|^p e^{- \frac{\alpha p |\sigma|^2}{2} }. \end{align}

The question is then whether or not the "sampling operator" mapping $F_\alpha ^p (\mathbb{C}^n)$ to $F_\alpha ^p (\mathbb{C}^n)$ given by \begin{align} f \mapsto \sum_{ \sigma \in \epsilon \mathbb{Z}^{2n} } f(\sigma) e^{ \alpha (z \cdot \overline{\sigma} ) - \alpha |\sigma|^2 } \nonumber \end{align} is invertible for small enough $\epsilon$.

Elementary Hilbert space theory says that for $p = 2$ it is, but I don't find where in the literature it is shown for $p \neq 2$, or shown not to be invertible. It's easy to show that this operator is injective and has dense range, but proving surjectivity seems to no easy task.

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I think the preprint may ask most of your question even in the polyanalytic setting.

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Hey Nelson thanks. Actually I carefully checked Grochenig's "Describing functions: atomic decomposition vs. frames" and the result can be proven pretty easily by using his theory of coorbit spaces (the proof is more or less entirely in that paper, though when specializing to the Fock space, one gets a much shorter proof.) Interestingly, it appeared that the above question answered in the affirmative was a well known "folklore" theorem. I have no idea if something stronger (i.e. precisely for what \epsilon it's true, or if it's true for other lattices) is also part of the folklore though. – Joshua Isralowitz Aug 18 '11 at 11:38

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