References of "Simons, Laurent"
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See detailFonction de Riemann généralisée
Simons, Laurent ULg; Bastin, Françoise ULg; Nicolay, Samuel ULg

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 ▲]

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See detailAbout the Multifractal Nature of Cantor's Bijection
Simons, Laurent ULg; Nicolay, Samuel ULg

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 ▲]

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See detailAn adaptation of $S^{\nu}$ spaces
Simons, Laurent ULg; Bastin, Françoise ULg; Nicolay, Samuel ULg

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 ▲]

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See detailRégularité de la fonction de Cantor
Simons, Laurent ULg; Nicolay, Samuel ULg

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 ▲]

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See detailAbout non stationary multiresolution analysis and wavelets
Bastin, Françoise ULg; Simons, Laurent ULg

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 ▲]

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See detailAbout the Regularity of Cantor's Bijection
Simons, Laurent ULg; Nicolay, Samuel ULg

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 ▲]

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See detailNon stationary Wavelets
Simons, Laurent ULg

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 ▲]

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See detailA note about non stationary multiresolution analysis
Simons, Laurent ULg

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 ▲]

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See detailNon Stationary Multiresolution Analysis
Simons, Laurent ULg

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 ▲]

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See detailLa fonction de Cantor
Simons, Laurent ULg

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.

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