About the Uniform Hölder Continuity of Generalized Riemann Function Bastin, Françoise ; Nicolay, Samuel ; Simons, Laurent in Mediterranean Journal of Mathematics (in press) In this paper, we study the uniform H\"{o}lder continuity of the generalized Riemann function~$R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac ... [more ▼] In this paper, we study the uniform H\"{o}lder continuity of the generalized Riemann function~$R_{\alpha,\beta}$ (with $\alpha>1$ and $\beta>0$) defined by \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x\in\mathbb{R}, \] using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of~$R_{\alpha,\beta}$ as $\beta$ tends to infinity. [less ▲] Detailed reference viewed: 16 (1 ULg)Topology on new sequence spaces defined with wavelet leaders Bastin, Françoise ; Esser, Céline ; Simons, Laurent E-print/Working paper (2015) Using wavelet leaders instead of wavelet coefficients, new sequence spaces of type Sν are defined and endowed with a natural topology. Some classical topological properties are then studied; in particular ... [more ▼] Using wavelet leaders instead of wavelet coefficients, new sequence spaces of type Sν are defined and endowed with a natural topology. Some classical topological properties are then studied; in particular, a generic result about the asymptotic repartition of the wavelet leaders of a sequence in Lν is obtained. Eventually, comparisons and links with Oscillation spaces are also presented as well as with Sν spaces. [less ▲] Detailed reference viewed: 8 (2 ULg)Fonction de Riemann généralisée Simons, Laurent ; Bastin, Françoise ; Nicolay, Samuel Conference (2014, September 22) Dans cet exposé, nous étudions la régularité de la fonction de Riemann généralisée~$R_{\alpha,\beta}$ (avec $\alpha>1$ et $\beta>0$) définie par \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n ... [more ▼] Dans cet exposé, nous étudions la régularité de la fonction de Riemann généralisée~$R_{\alpha,\beta}$ (avec $\alpha>1$ et $\beta>0$) définie par \[ R_{\alpha,\beta}(x)=\sum_{n=1}^{+\infty}\frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x\in\R. \] En particulier, nous déterminons son exposant de Hölder uniforme. Pour terminer, nous analysons le comportement de~$R_{\alpha,\beta}$ lorsque le paramètre $\alpha$ ou $\beta$ tend vers l'infini. Cet exposé est basé sur un travail en collaboration avec F. Bastin et S. Nicolay. [less ▲] Detailed reference viewed: 16 (4 ULg)Detection of non concave and non increasing multifractal spectra using wavelet leaders (Part I) Esser, Céline ; Kleyntssens, Thomas ; Bastin, Françoise et al Conference (2014, May 22) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdor ff dimension of the set of points which have the same H ölder exponent. For real-life signals, the computation of the spectrum of singularities from its de finition is not feasible. Multifractal formalisms are used to approximate this spectrum. Currently, there exist several methods. In this talk, we present a new multifractal formalism based on the wavelet leaders of a signal which allows to detect non concave and non increasing spectra. [less ▲] Detailed reference viewed: 26 (6 ULg)A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I) Esser, Céline ; Kleyntssens, Thomas ; Nicolay, Samuel et al Conference (2014, March 25) Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to ... [more ▼] Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdorff dimension of the set of points which have the same Hölder exponent. The spectrum of singularities of many mathematical functions can be determined directly from its definition. However, for many real-life signals, the numerical determination of their Hölder regularity is not feasible. Therefore, one cannot expect to have a direct access to their spectrum of singularities and one has to find an indirect way to compute it. A multifractal formalism is a formula which is expected to yield the spectrum of singularities from quantities which are numerically computable. Several multifractal formalisms based on the wavelet coefficients of a signal have been proposed to estimate its spectrum. The most widespread of these formulas is the so-called thermodynamic multifractal formalism, based on the Frish-Parisi conjecture. This formalism presents two drawbacks: it can hold only for spectra that are concave and it can yield only the increasing part of the spectrum. This first problem can be avoided using Snu spaces. The second one can be avoided using a formalism based on wavelet leaders of the signal. In this talk, we propose a new multifractal formalism, based on a generalization of the Snu spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. An implementation of this method is presented in the talk "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)" of T. Kleyntssens. [less ▲] Detailed reference viewed: 42 (11 ULg)Algebrability and nowhere Gevrey differentiability Bastin, Françoise ; ; Esser, Céline et al in Israel Journal of Mathematics (2014) We show that there exist c-generated algebras (and dense in C^infty([0,1])) every nonzero element of which is a nowhere Gevrey diff erentiable function. This leads to results of dense algebrability (and ... [more ▼] We show that there exist c-generated algebras (and dense in C^infty([0,1])) every nonzero element of which is a nowhere Gevrey diff erentiable function. This leads to results of dense algebrability (and, therefore, lineability) of functions enjoying this property. In the process of proving these results we also provide a new construction of nowhere Gevrey di fferentiable functions. [less ▲] Detailed reference viewed: 83 (33 ULg)Large deviation spectra based on wavelet leaders Bastin, Françoise ; Esser, Céline ; Jaffard, Stéphane E-print/Working paper (2014) We introduce a new multifractal formalism, based on distributions of wavelet leaders, which allows to detect non-concave and decreasing multifractal spectra, and we investigate the properties of the ... [more ▼] We introduce a new multifractal formalism, based on distributions of wavelet leaders, which allows to detect non-concave and decreasing multifractal spectra, and we investigate the properties of the associated function spaces. [less ▲] Detailed reference viewed: 6 (2 ULg)An adaptation of $S^{\nu}$ spaces Simons, Laurent ; Bastin, Françoise ; Nicolay, Samuel Scientific conference (2013, May 31) The $S^\nu$ spaces have been introduced in 2004 by S. Jaffard in the context of multifractal analysis. In comparison with Besov spaces (the classical functional setting to study signals), these spaces of ... [more ▼] The $S^\nu$ spaces have been introduced in 2004 by S. Jaffard in the context of multifractal analysis. In comparison with Besov spaces (the classical functional setting to study signals), these spaces of functions related to the distribution of wavelet coefficients allow to obtain more information on the Hölder regularity of a signal. From a point of view of functional analysis, the $S^nu$ spaces can be considered as sequence spaces (because they are robust). Some properties (topology, complete metric, $p$-locally convexity,...) have been studied. The purpose of the talk is to present the beginning of an adaptation of the $S^nu$ spaces when the discrete wavelet coefficients are replaced by continuous wavelet transform coefficients. [less ▲] Detailed reference viewed: 32 (9 ULg)About non stationary multiresolution analysis and wavelets Bastin, Françoise ; Simons, Laurent in Results in Mathematics (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 ▲] Detailed reference viewed: 81 (22 ULg)About Generic Properties of "Nowhere Analyticity" Bastin, Françoise ; Nicolay, Samuel ; Esser, Céline Conference (2012, May 08) A infinitely differentiable function f is is analytic at a point x if its Taylor series at this point converges to f on an open neighbourhood of x; if this is not the case, f has a singularity at x. A ... [more ▼] A infinitely differentiable function f is is analytic at a point x if its Taylor series at this point converges to f on an open neighbourhood of x; if this is not the case, f has a singularity at x. A function with a singularity at each point of the interval is called nowhere analytic on the interval. In this talk, we show that the set of nowhere analytic functions is prevalent in the Frechet space C([0;1]). We get then a deeper result using Gevrey classes. [less ▲] Detailed reference viewed: 39 (20 ULg)Prevalence of ''nowhere analyticity'' Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel 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. Detailed reference viewed: 54 (23 ULg)Prevalence of ''nowhere analyticity'' Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel 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. Detailed reference viewed: 54 (23 ULg)Prevalence of ''nowhere analyticity'' Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel 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. Detailed reference viewed: 54 (23 ULg)Bastin, Françoise ; Esser, Céline ; Nicolay, Samuel in Studia Mathematica (2012), 210(3), A walk from multifractal analysis to functional analysis with S\nu, and back Aubry, Jean-Marie ; Bastin, Françoise in Barral J; Seuret S. (Ed.) Proceedings of ''Fractals and Related Fields'', Monastir, September 2007 (2010) With the S\nu spaces introduced by Jaffard in the context of multifractal analysis to extend the Besov spaces environment, functional analysis received a gift from concrete applications. These spaces led ... [more ▼] With the S\nu spaces introduced by Jaffard in the context of multifractal analysis to extend the Besov spaces environment, functional analysis received a gift from concrete applications. These spaces led to new results in multifractal analysis, but also brought concrete objects to study as new examples by the typical various tools and aspects of functional analysis, with hope to provide some new points of view from which to consider multifractal analysis questions. [less ▲] Detailed reference viewed: 58 (26 ULg)Diametral Dimension of some pseudoconvex multiscale spaces Aubry, Jean-Marie ; Bastin, Françoise 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 ▲] Detailed reference viewed: 71 (31 ULg)Advanced topology on the multiscale sequence spaces S-nu Aubry, Jean-Marie ; Bastin, Françoise in Journal of Mathematical Analysis & Applications (2009), 350(2), 439-454 We put-sue the Study of the multiscale spaces S-v introduced by Jaffard in the context of multifractal analysis. We give the necessary and Sufficient condition for S-v to be locally p-convex, and exhibit ... [more ▼] We put-sue the Study of the multiscale spaces S-v introduced by Jaffard in the context of multifractal analysis. We give the necessary and Sufficient condition for S-v to be locally p-convex, and exhibit a sequence of p-norms that defines its natural topology. The strong topological dual of S-v is identified to another sequence space depending on v, endowed with an inductive limit topology. As a particular case, we describe the dual of a countable intersection of Besov spaces. (C) 2007 Elsevier Inc. All rights reserved. [less ▲] Detailed reference viewed: 85 (12 ULg)Prevalenee of multifractal functions in S-nu spaces Aubry, Jean-Marie ; Bastin, Françoise ; 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 ▲] Detailed reference viewed: 55 (27 ULg)The S\nu spaces: new spaces defined with wavelet coefficients and related to multifractal analysis Aubry, Jean-Marie ; Bastin, Françoise ; et al in International Journal of Applied Mathematics & Statistics (2007), 7(Fe07), 82-95 In the context of multifractal analysis, more precisely in the context of the study of H\"older regularity, Stéphane Jaffard introduced new spaces of functions related to the distributionof wavelet ... [more ▼] In the context of multifractal analysis, more precisely in the context of the study of H\"older regularity, Stéphane Jaffard introduced new spaces of functions related to the distributionof wavelet coefficients, the ${\cal S}^{\nu}$ spaces. From a functional analysis point of view, one can define the corresponding sequence spaces, endow them with natural topologies and study their properties. The results lead to construct probability Borel measures with applications in the context of multifractal analysis. [less ▲] Detailed reference viewed: 117 (28 ULg)Topological properties of the sequence spaces S-nu Aubry, Jean-Marie ; Bastin, Françoise ; et al 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 ▲] Detailed reference viewed: 61 (24 ULg) |
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