Variantes sur un th\'eor\`eme de Cand\`es, Romberg et Tao

Abstract

Variations on a theorem of Cand\`es, Romberg and Tao The CRT theorem reconstructs a signal from a sparse set of frequencies, a paradigm of Compressed sensing. The signal is assumed to be carried by a small number of points, s, in a large cyclic set, of order N; the frequencies consist of C s log N points chosen randomly in Z/N Z; the reconstruction is based on a minimal extrapolation in the Wiener algebra of Z/N Z of the restriction of the Fourier transform of the signal to the chosen set of frequencies. The probability of reconstructing the signal is nearly 1 when C is large. The statement should be modified when we want all signals carried by s points to be reconstructed in that way. The CRT approach is based on random matrices, here the approach is classical Fourier analysis.