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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: 61 (28 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: 82 (38 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: 87 (13 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: 57 (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: 122 (32 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: 77 (36 ULg) |
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