Unconditional basis in Bargmann space -- new proof of Gröchenig-Walnut theorem (Q2706842)

The Bargmann space \(A^p(\mathbb C)\) is defined as follows: NEWLINE\[NEWLINEA^p(\mathbb C)=\biggl\{ f \text{ entire}:\int_{\mathbb C}|f(z)e^{-\pi|z|^2/2}|^p d\lambda(z) <\infty \biggr\},\quad 1\leq p<\infty, NEWLINE\]NEWLINE endowed with the natural norm NEWLINE\[NEWLINE \|f\|_{A^p(\mathbb C)}=\biggl(\int_{\mathbb C}|f(z) e^{-\pi|z|^2/2}|^p d\lambda(z)\biggr)^{1/p}. NEWLINE\]NEWLINE In this paper a new and short proof for the Gröchenig-Walnut theorem giving an unconditional basis in \(A^p(\mathbb C)\) is shown. This proof omits the use of Zak transform exploited in the original proof.