Baire generic histograms of wavelet coefficients and large deviation formalism in Besov and Sobolev spaces

Abstract

Histograms of wavelet coefficients are expressed in terms of the wavelet profile and the wavelet density. The large deviation multifractal formalism states that if a function f has a minimal uniform Hölder regularity then its Hölder spectrum is equal to the wavelet density. The purpose of this paper is twofold. Firstly, we compute generically (in the sense of Baire's categories) these histograms in Besov B p s , q ( T ) and L p , s ( T ) spaces, where T is the torus R d / Z d (resp. in the Baire's vector space V = ⋂ ε > 0 , p > 0 B p s ( 1 p ) − ε p , p where s : q ↦ s ( q ) is a C 1 and concave function on R + satisfying 0 ⩽ s ′ ⩽ d and s ( 0 ) > 0 ). Secondly, as an application, we deduce some extra generic properties for the histograms in these spaces, and study the generic validity of the large deviation multifractal formalism in Besov and L p , s spaces for s > d / p (resp. in the above space V).