Zak transform characterization of S 0

We present a characterization of the modulation space S0 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 % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey % icI48aaKGeaeaacaaIWaGaaiilamaalaaabaGaaG4maaqaaiaaikda % aaaacaGLBbGaayzkaaaaaa!3DF1! $$\lambda \in \left[ {0,\frac{3}{2}} \right)$$ . 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 Grochenig 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.