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About the Uniform Hölder Continuity of Generalized Riemann Function Bastin, Françoise ; Nicolay, Samuel ; Simons, Laurent in Mediterranean Journal of Mathematics (2016), 13(1), 101-117 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: 47 (15 ULg)Hölder Continuity and Wavelets Simons, Laurent Doctoral thesis (2015) There exist a lot of continuous nowhere differentiable functions, but these functions do not have the same irregularity. Hölder continuity, and more precisely Hölder exponent, allow to quantify this ... [more ▼] There exist a lot of continuous nowhere differentiable functions, but these functions do not have the same irregularity. Hölder continuity, and more precisely Hölder exponent, allow to quantify this irregularity. If the Hölder exponent of a function takes several values, the function is said multifractal. In the first part of this thesis, we study in details the regularity and the multifractality of some functions: the Darboux function, the Cantor bijection and a generalization of the Riemann function. The theory of wavelets notably provides a tool to investigate the Hölder continuity of a function. Wavelets also take part in other contexts. In the second part of this thesis, we consider a nonstationary version of the classical theory of wavelets. More precisely, we study the nonstationary orthonormal bases of wavelets and their construction from a nonstationary multiresolution analysis. We also present the nonstationary continuous wavelet transform. For some irregular functions, it is difficult to determine its Hölder exponent at each point. In order to get some information about this one, new function spaces based on wavelet leaders have been introduced. In the third and last part of this thesis, we present these new spaces and their first properties. We also define a natural topology on them and we study some properties. [less ▲] Detailed reference viewed: 88 (27 ULg)About the Regularity of Cantor's Bijection Simons, Laurent ; Nicolay, Samuel Conference (2015, May 25) In 1878, Cantor proved that there exists a one-to-one correspondence between the points of the unit line segment [0,1] and the points of the unit square [0,1]². Since this application is defined via ... [more ▼] In 1878, Cantor proved that there exists a one-to-one correspondence between the points of the unit line segment [0,1] and the points of the unit square [0,1]². Since this application is defined via continued fractions, it is very hard to have any intuition about its smoothness. In this talk, we explore the regularity and the fractal nature of Cantor's bijection, using some notions concerning the metric theory and the ergodic theory of continued fractions. This talk is based on a joint work with S. Nicolay. [less ▲] Detailed reference viewed: 19 (0 ULg)Topology on new sequence spaces defined with wavelet leaders Bastin, Françoise ; Esser, Céline ; Simons, Laurent in Journal of Mathematical Analysis and Applications (2015), 431(1), 317-341 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: 52 (20 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: 34 (6 ULg)About the Multifractal Nature of Cantor's Bijection Simons, Laurent ; Nicolay, Samuel Conference (2014, March 25) In this talk, we present the Cantor's bijection between the irrational numbers of the unit interval [0,1] and the irrational numbers of the unit square [0,1]². We explore the regularity and the fractal ... [more ▼] In this talk, we present the Cantor's bijection between the irrational numbers of the unit interval [0,1] and the irrational numbers of the unit square [0,1]². We explore the regularity and the fractal nature of this map. This talk is based on a joint work with S. Nicolay. [less ▲] Detailed reference viewed: 37 (4 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: 36 (9 ULg)Régularité de la fonction de Cantor Simons, Laurent ; Nicolay, Samuel Scientific conference (2013, January 28) La fonction de Cantor, bijection entre $[0,1]$ et $[0,1]^2$, est définie via les fractions continues. Par conséquent, il est assez difficile d'avoir une quelconque intuition sur son comportement. Le but ... [more ▼] La fonction de Cantor, bijection entre $[0,1]$ et $[0,1]^2$, est définie via les fractions continues. Par conséquent, il est assez difficile d'avoir une quelconque intuition sur son comportement. Le but de cet exposé est de présenter cette fonction particulière ainsi que sa régularité (continuité et régularité höldérienne). [less ▲] Detailed reference viewed: 28 (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: 104 (32 ULg)About the Regularity of Cantor's Bijection Simons, Laurent ; Nicolay, Samuel Conference (2012, May 07) Multifractal analysis has been introduced in the context of turbulence. Some tools have been developed to study the solutions of some PDEs. In this talk, we will examine the regularity of Cantor's ... [more ▼] Multifractal analysis has been introduced in the context of turbulence. Some tools have been developed to study the solutions of some PDEs. In this talk, we will examine the regularity of Cantor's bijection between the irrational numbers of the unit interval [0,1] and the irrational numbers of the unit square [0,1]^2. We will particularly show that its H older exponent is equal to 1/2 almost everywhere (with respect to the Lebesgue measure). [less ▲] Detailed reference viewed: 26 (6 ULg)Non stationary Wavelets Simons, Laurent Scientific conference (2011, December 21) In the presentation, I first compare the construction of orthonormal bases of wavelets from a multiresolution in the stationary and the non stationary case. Then, I expose some generalizations of ... [more ▼] In the presentation, I first compare the construction of orthonormal bases of wavelets from a multiresolution in the stationary and the non stationary case. Then, I expose some generalizations of characterizations of orthonormal bases of wavelets in the non stationary case. Finally, I speak about the non stationary case for the continuous wavelet transform. [less ▲] Detailed reference viewed: 35 (5 ULg)A note about non stationary multiresolution analysis Simons, Laurent Conference (2011, July 28) An orthonormal basis of wavelets of $L^2(\mathbb{R})$ is an orthonormal basis of $L^2(\mathbb{R})$ of type \[ \psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\mathbb{Z}. \] A classical method to obtain ... [more ▼] An orthonormal basis of wavelets of $L^2(\mathbb{R})$ is an orthonormal basis of $L^2(\mathbb{R})$ of type \[ \psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\mathbb{Z}. \] A classical method to obtain such bases consists in constructing a multiresolution analysis. When the mother wavelet $\psi$ depends on the scale (i.e. the index $j$), a non stationary version of multiresolution analysis is then used. We generalize different characterizations of orthonormal bases of wavelets to the non stationary case (as main reference for the stationary case, we used results presented in "A First Course of Wavelets" of E. Hernandez and G. Weiss). [less ▲] Detailed reference viewed: 30 (10 ULg)Non Stationary Multiresolution Analysis Simons, Laurent Poster (2010, September 13) An orthonormal basis of wavelets of $L^2(\R)$ is an orthonormal basis of $L^2(\R)$ of type \[ \psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\Z. \] A classical method to obtain such bases consists in ... [more ▼] An orthonormal basis of wavelets of $L^2(\R)$ is an orthonormal basis of $L^2(\R)$ of type \[ \psi_{j,k}=2^{j/2}\psi(2^j\cdot-k),\quad j,k\in\Z. \] A classical method to obtain such bases consists in constructing a multiresolution analysis. When the mother wavelet $\psi$ depends on the scale (i.e. the index $j$), a non stationary version of multiresolution analysis is then used. It is for example the case in the general context of Sobolev spaces. We generalize different characterizations in the standard theory of wavelets to the case of multi-scales wavelets and non stationary multiresolution analyses. [less ▲] Detailed reference viewed: 53 (18 ULg)La fonction de Cantor Simons, Laurent Master's dissertation (2009) Ce mémoire est une introduction à l'étude de la bijection de Cantor, bijection entre l'intervalle unité et le carré unité. Detailed reference viewed: 105 (21 ULg) |
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