Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces

Abstract

We obtain estimates for the Lp-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of R2, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions T×W of the time-frequency plane using the Fourier transform to intertwine time and frequency, we localize the representation of the signals in terms of the short-time Fourier transform in sets Δ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for f∈L2(R) from its STFT values on arbitrary discs.