Colloquium de l'IDPQuelques problèmes mathématiques posés par le théorème de Shannon
Thursday 14 June 2012 14:00 - Orléans - Amphithéâtre S
Let us assume that a function $F\in L^2(\R^n)$ has a ``small support". More precisely we are given a small number $\beta$ and we assume that $F$ is supported by a compact set $K$ whose measure $|K|$ does not exceed $\beta.$ The problem which is addressed today is to take advantage of this property in order to sample efficiently the Fourier transform of $F.$ Following the paradigm of ``compressed sensing" the sampling rate should match the sparsity of the signal (or of the image). Our main Theorem gives a solution to this problem. A new class of frames (wealthy frames) are defined and studied in order to prove this theorem. Let $f$ be the inverse Fourier transform of $F.$ Then $f$ is a generalized band-limited function. We will prove that an efficient sampling of $F$ is obtained by $f(\lambda),\,\lambda \in \Lambda,$ where $\Lambda$ is a simple quasicrystal whose density $d$ satisfies $d>\beta.$ This result is sharp.