The multifractal formalism for functions relates some functional norms of a signal to its "Hölder spectrum" (which is the dimension set points where has given Hölder regularity). This was initially introduced by Frisch and Parisi in order numerically determine spectrum fully turbulent fluids; it later extended Arneodo, Bacry, Muzy using wavelet techniques since been used many physicists. Until now, only supported heuristic arguments verified few specific examples. Our purpose investigate...

Following a basic idea of Wilson [“Generalized Wannier functions,” preprint] orthonormal bases for $L^2 (\mathbb{R})$ which are variation on the Gabor scheme constructed. More precisely, $\phi \in L^2 is constructed such that $\psi _{ln } $, $l \mathbb{N}$, $n \mathbb{Z}$, defined by \[ \begin{gathered} \psi _{0n} (x) = \phi \left( {x - n} \right) \hfill \\ _{in} \sqrt 2 \frac{n} {2}} \right)\cos (2\pi lx)\,{\text{if}}\,l \ne 0,\,l + n 2\mathbb{Z} \right)\sin 1, \end{gathered} \] constitute...

This paper shows that the use of wavelets to discretize an elliptic problem with Dirichlet or Neumann boundary conditions has two advantages: explicit diagonal preconditioning makes condition number corresponding matrix become bounded by a constant and order approximation is locally spectral type (in contrast classical methods); using conjugate gradient method, one thus obtains fast numerical algorithms resolution. A comparison also drawn between wavelet methods.

In this paper, we shall compare three notions of pointwise smoothness: the usual definition, J.M. Bony's two-microlocal spaces Cxós,, and corresponding definition on wavelet coefficients .The purpose is mainly to show that these provide "good substitutes" for Hdlder regularity condition ; they can be very precisely compared with condition, have more functional properties, characterized by conditions coefficients.We also give applications properties.In Part 2 some results microlocal contained...

In this paper we introduce and study the self-similar functions. We prove that these functions have a concave spectrum (increasing then decreasing) different formulas were proposed for multifractal formalism allow us to determine either whole increasing part of their or it. One methods (the wavelet-maxima method) yields complete spectrum.

We determine the Holder regularity of Riemann's function at each point; we deduce from this analysis its spectrum singularities, thus showing multifractal nature.

We prove that the Hölder singularities of random lacunary wavelet series are chirps located on fractal sets. determine Hausdorff dimensions these singularities, and a.e. modulus continuity series. Lacunary thus turn out to be a new example multifractal functions.

Textures in images can often be well modeled using self-similar processes while they may simultaneously display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will first shown that accurate joint estimates the parameters are performed replacing standard 2D-discrete wavelet transform with hyperbolic transform, which permits use different dilation...