Coherent states, Gabor multipliers, and the Banach Gelfand Triple
Hans G. Feichtinger (NuHAG, Faculty of Mathematics, Univ. Vienna)
given at Marrakesch (03.04.19)
From the point of view of Abstract Harmonic Analysis (AHA) all we need for Time-Frequency Analyis (TFA) is a locally compact Abelian (LCA) group, which allows us to define the Short-Time or Sliding Window Fourier Transform (STFT) on the Hilbert space L^2(G).
For the case of the Euclidean group G = R^d one can take the Gauss-function as a specific ``window'' or (thinking of the inversion formula, which allows to rebuild the given
function f in L^2(G) from the STFT) the ``atom'', i.e. write it as a (continuous) superposition of time-frequency shifted copies of the atom, the so-called coherent states. Gabor Analysis goes one step further, by replacing the continuous integral by an infinite sum. Anti-Wick operators (we also call them STFT-multipliers) are obtained by multiplying the STFT with some function before resynthesis, and in a similar way Gabor multipliers are obtained by multiplying the Gabor coefficients (indexed by some lattice in phase space) by some weight factors, before doing synthesis.
The goal of this talk will be to show how results about these operators can be compactly described using the Banach Gelfand triple (S_0,L^2,S_0'), also known ad modulation spaces
(M1,L^2,M^\infty). We view the Segal algebra S_0 as a Fourier invariant Banach algebra of test functions, and its dual as a space of ``mild'' distributions. They can be introduced making use only of Riemann integrals, without reference to Lebesgue measures and topological vector spaces.
The talk will focus on the usefulness of this perspective on questions in TFA, also indicating how this ``rigged Hilbert space'' situation allows to derive robustness properties of e.g. the Gabor frames and Gabor multipliers. Such results are not possible of one limits the view on the pure Hilbert space setting. Even more importantly one can ask whether the TFA and Gabor Analysis that can be realized over finite Abelian groups (and which can be implemented in a MATLAB environment) can be used to reliably approximate their continuous counterpart. Among others we will reveal an approach to the realization of discrete Hermite functions, based on experimental insight. As a matter of fact it coincides with a method proposed not too long ago by N. Cotfas. It is also the basis for what we consider to be an appropriate implementation of the fractional Fourier transform. The theoretical claims will be illustrated by numerical results.