Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Non-linear approximation and Muckenhoupt weights.


Code(s) de Classification MSC:

Résumé: In a completely general atomic setting of an unconditional basis in a (quasi-) Banach space we show that representation of the spaces of $m$-terms approximation as Lorentz spaces is equivalent to the verification of Jackson and Bernstein inequalities, and that either of these properties is equivalent to the Temlyakov property. The proof is very direct, and especially does not use interpolation theory. We apply this result to establish a representation theorem when the norm of the (quasi-) Banach space is given by a quadratic variation formula and thanks to a condition called $p$-reverse inequality. This quadratic variation framework is in fact very rich and contains as examples the cases of Hardy spaces. We also consider the cases of 'weighted' Hardy and Lebesgue spaces when the weight belongs to a Muckenhoupt class and the basis is a wavelet basis. We finally consider the case of warped wavelets, which appear as composition of a usual wavelet with a scaling function, and behave, under appropriate conditions on the scaling as well as ordinary wavelets. This provides a new example of bases well adapted to approximation.

Mots Clés: m-term approximation ; wavelet bases ; Muckenhoupt weight ; Lorentz spaces

Date: 2002-01-31

Prépublication numéro: PMA-704