About non stationary multiresolution analysis and wavelets

in Results in Mathematics [=RM] (2013), 63(1), 485-500

The characterization of orthonormal bases of wavelets by means of convergent series involving only the mother wavelet is known, as well as the characterization of wavelets which can be constructed from a ... [more ▼]

The characterization of orthonormal bases of wavelets by means of convergent series involving only the mother wavelet is known, as well as the characterization of wavelets which can be constructed from a stationary multiresolution analysis or a scaling function (see for example the book of Hernandez-Weiss and references therein). Here we show that under some asymptotic condition, these results remain true in the non stationary case. [less ▲]

Prevalence of ''nowhere analyticity''

in Studia Mathematica (2012), 210(3),

This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study in Gevrey classes of functions.

Diametral Dimension of some pseudoconvex multiscale spaces

in Studia Mathematica (2010), 197(1), 27-42

Stemming from the study of signals via wavelet coefficients, the spaces S\nu are complete metrizable and separable topological vector spaces, parametrized by a function \nu, whose elements are sequences ... [more ▼]

Stemming from the study of signals via wavelet coefficients, the spaces S\nu are complete metrizable and separable topological vector spaces, parametrized by a function \nu, whose elements are sequences indexed by a binary tree. Several papers were devoted to their basic topology; recently it was also shown that depending on \nu, S\nu may be locally convex, locally p-convex for some p > 0, or not at all, but under a minor condition they are always pseudoconvex. We tackle here some more sophisticated properties: their diametral dimensions show that they are Schwartz but not nuclear spaces. Moreover, Ligaud’s example of a Schwartz pseudoconvex non p-convex space is actually a particular case of S\nu. [less ▲]

Prevalenee of multifractal functions in S-nu spaces

in Journal of Fourier Analysis and Applications (2007), 13(2), 175-185

Spaces called S-v were introduced by Jaffard [16] as spaces of functions characterized by the number similar or equal to 2(v(alpha)j) of their wavelet coefficients having a size greater than or similar to ... [more ▼]

Spaces called S-v were introduced by Jaffard [16] as spaces of functions characterized by the number similar or equal to 2(v(alpha)j) of their wavelet coefficients having a size greater than or similar to 2(-alpha j) at scale j. They are Polish vector spaces for a natural distance. In those spaces we show that multifractal functions are prevalent (an infinite-dimensional "almost-every"). Their spectrum of singularities can be computed from v, which justifies a new multifractal formalism, not limited to concave spectra. [less ▲]

Topological properties of the sequence spaces S-nu

in Journal of Mathematical Analysis and Applications (2006), 321(1), 364-387

We define sequence spaces based on the distributions of the wavelet coefficients in the spirit of [S. Jaffard, Beyond Besov spaces, part I: Distributions of wavelet coefficients, J. Fourier Anal. Appl. 10 ... [more ▼]

We define sequence spaces based on the distributions of the wavelet coefficients in the spirit of [S. Jaffard, Beyond Besov spaces, part I: Distributions of wavelet coefficients, J. Fourier Anal. Appl. 10 (2004) 221-246]. We study their topology and especially show that they can be endowed with a (unique) complete metric for which compact sets can be explicitly described and we study properties of this metric. We also give relationships with Besov spaces. (c) 2005 Elsevier Inc. All rights reserved. [less ▲]

Deficient splines and wavelets

in Revista Ciencas Matematicas (Universidad de la Habana) (2005)

Coherent states, wavelets and applications G24 - LLN

in Group 24 : Physical and Mathematical Aspects of Symmetries (2003)

A general recurrence relation between the moments of a scaling function

in Group 24 : Physical and Mathematical Aspects of Symmetries (2003)

Under natural and weak hypotheses, we prove a reproducing formula for polynomials. Then we obtain a new recurrence relation between the moments of a scaling function and a new exact formula for the ... [more ▼]

Under natural and weak hypotheses, we prove a reproducing formula for polynomials. Then we obtain a new recurrence relation between the moments of a scaling function and a new exact formula for the computation of moments of even order. [less ▲]

Spline wavelets in periodic Sobolev spaces and application to high order collocation methods

in Revista de la Union Matematica Argentina (2003), 44(1), 53-74

In this paper, we present a particular family of spline wavelets constructed from the Chui-Wang Riesz basis of $L^2(\mathbb{R})$. The construction is explicit, allowing the study of specific functional ... [more ▼]

In this paper, we present a particular family of spline wavelets constructed from the Chui-Wang Riesz basis of $L^2(\mathbb{R})$. The construction is explicit, allowing the study of specific functional properties and rather easy handling in numerical computations. This family constitutes a Riesz hierarchical basis in periodic Sobolev spaces. We also present a necessary and sufficient condition of strong ellipticity for pseudodifferential operators obtained with respect to these splines. It uses a new expression for the numerical symbol of the boundary integral operators. This expression allows us to use efficiently collocation methods with different meshes and splines. [less ▲]

Quintic deficient spline wavelets

in Bulletin de la Société Royale des Sciences de Liège (2002), 71(3), 121-144

We show explicitely how to construct scaling functions and wavelets using quintic deficient splines with compact support and symmetry properties.