Nonlinear approximation with dictionaries. II. Inverse estimates

Rémi Gribonval 1 Morten Nielsen 2
1 METISS - Speech and sound data modeling and processing
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : In this paper, which is the sequel to [R. Gribonval and M. Nielsen. Nonlinear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl., 10(1):51-71, 2004], we study inverse esti- mates of the Bernstein type for nonlinear approximation with structured redun- dant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequal- ity obtained for such dictionaries is proved to be sharp, but it has an exponent that does not match that of the corresponding Jackson inequality.
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Rémi Gribonval, Morten Nielsen. Nonlinear approximation with dictionaries. II. Inverse estimates. Constructive Approximation, Springer Verlag, 2006, 24 (2), pp.157--173. ⟨10.1007/s00365-005-0621-x⟩. ⟨inria-00544906⟩

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