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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 41A25 Rate of convergence, degree of approximation
- 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
- 65F99 None of the above but in this section
- 65N12 Stability and convergence of numerical methods
- 65N55 Multigrid methods; domain decomposition

**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*