Wiener Estimates on Modulation Spaces

Abstract

We characterise modulation spaces by suitable Wiener estimates on the short-time Fourier transforms of the involved functions and distributions. We use the results to refine some formulae on periodic distributions with Lebesgue estimates on their coefficients.1991 Mathematics Subject Classification. Primary: 42C20, 43A32, 42B35, 46E10, Secondary: 46A16, 35A22, 37A05, 46E35.

The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

“…The following result is essential when characterizing elements in modulation spaces in terms of estimates of their Zak transforms. We omit the proof since the result is a consequence of [39,Proposition 2.6].…”

“…For example we prove that Z 1 is continuous and bijective from S (R d ) to the set of all quasi-periodic distributions on Q d, 1 . We also use some recent results in [39] on Wiener estimates to deduce different versions of the latter characterization. For example we show that (0.2) in combination with results in [39,Section 2] give…”

“…We also use some recent results in [39] on Wiener estimates to deduce different versions of the latter characterization. For example we show that (0.2) in combination with results in [39,Section 2] give…”