Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Répondant à une question posée par H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52] sur le problème de l'échantillonnage irrégulier des fonctions “band-limited”, nous prouvons que les quasicristaux sont des ensembles d'échantillonnage stable. Pour citer cet article : B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The famous Nyquist-Shannon sampling theorem has been recently improved by A. Olevskii and A. Ulanovskii. The present contribution is aimed at bridging the gap between their advance on irregular sampling and what is named "compressed sensing" in signal processing.

We introduce new multiscale representations for images which incorporate a specific geometric treatment of edges. The associated transforms are inherently nonlinear and nontensor product, in contrast to classical wavelet basis decompositions over which they exhibit visual improvement in terms of compression. This approach can be viewed as a bridge between edge detection and the nonlinear multiresolution representations of Ami Harten (see Journal of Applied Numerical Mathematics, vol.12, p.153-93, 1993).