Zak transform characterization of S₀

We present a characterization of the modulation space S_o in terms of the Zak transform of its elements. We illustrate our result by considering S^-λh, where h is the standard Gaussian, S is the "frame" operator corresponding to the critical-density Gabor system (h, a = 1, b = 1), and λ ∈ [0,3/2). Both the proof of the main result and the example require basics from Gabor frame theory; these are developed in a separate section. We further use a result from recent work by Gröchenig and Leinert on Wiener-type theorems in a non-commutative setting. We also present an extension of our main result to more general modulation spaces.