Discrete and finite approximations in Gabor Analysis
Hans G. Feichtinger
given at NuHAG seminar (10.03.10 11:15)
Gabor Analysis has the advantage of being available for arbitrary LCA groups, including the Euclidean Space as well as finite groups. Gabor Analysis has close relationships to Fourier analysis, hence Gabor analysis over finite groups can be efficiently realized using FFT-based algorithms (to determine the sampled STFT, doing Gabor synthesis, determining dual or tight Gabor atoms).
We will discuss a function spaces setting (using heavily the Segal algebra S_o(G) as a universal tool, but also SO' to describe sequences as discrete measures resp. distributions...) to come up with results of the following type:
Given a problem (such as determining the continuous Fourier transform), some norm (e.g. the L2 or SO-norm) and some $vareps > 0$ how can we efficiently/effectively determine within MATLAB (or some other computer program, doing a finite sequence of multiplications and logical steps) a (typically linear) procedure which allows to satisfy the given request of allowing to guarantee that the given error margins (in the chosen norm, and for the given $ varesp > 0$) are met. Of course one would like to know even that these are at least sub-optimal (in the sense of close to optimal) stratgies, in the sense of computational and storage cost.
This few-point is closer to implementation than the usual way of considering constructive versus non-constructive approaches to such problems (in the sense of constructive approximation theory).