Reference : Regularity of functions: Genericity and multifractal analysis
Dissertations and theses : Doctoral thesis
Physical, chemical, mathematical & earth Sciences : Mathematics
Regularity of functions: Genericity and multifractal analysis
Esser, Céline mailto [Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes >]
Université de Liège, ​Liège, ​​Belgique
Doctorat en Sciences
Bastin, Françoise mailto
Schneiders, Jean-Pierre mailto
Nicolay, Samuel mailto
Bonet, José mailto
Jaffard, Stéphane mailto
Seoane-Sepúlveda, Juan B. mailto
Wengenroth, Jochen mailto
[en] genericity ; multifractal formalism ; ultradifferentiable functions ; wavelet leaders
[en] As surprising as it may seem, there exist functions of C∞(R) which are nowhere analytic. When such an unexpected object is found, a natural question is to ask whether many similar ones may exist. A classical technique is to use the Baire category theorem and the notion of residuality. This notion is purely topological and does not give any information about the measure of the set of objects satisfying such a property. In this
purpose, the notion of prevalence has been introduced. Moreover, one could also wonder whether large algebraic
structures of such objects can be constructed. This question is formalized by the notion of lineability.
The first objective of the thesis is to go further into the study of nowhere analytic functions. It is known that the set of nowhere analytic functions is residual and lineable in C∞([0, 1]). We prove that the set of nowhere analytic functions is also prevalent in C∞([0, 1]). Those results of genericity are then generalized using Gevrey classes, which can be seen as intermediate between the space of analytic functions and the space of infinitely differentiable functions. We also study how far such results of genericity could be extended to spaces of ultradifferentiable functions, defined using weight sequences or using weight functions.
The second main objective is to study the pointwise regularity of functions via their multifractal spectrum. Computing the multifractal spectrum of a function using directly its definition is an unattainable goal in most of the practical cases, but there exist heuristic methods, called multifractal formalisms, which allow to estimate this spectrum and which give satisfactory results in many situations. The Frisch-Parisi conjecture, classically used and based on Besov spaces, presents two disadvantages: it can only hold for spectra that are concave and
it can only yield the increasing part of spectra. Concerning the first problem, the use of Snu spaces allows to deal with non-concave increasing spectra. Concerning the second problem, a generalization of the Frisch-Parisi conjecture obtained by replacing the role played by wavelet coefficients by wavelet leaders allows to recover the decreasing part of concave spectra.
Our purpose in this thesis is to combine both approaches and define a new formalism derived from large deviations based on statistics of wavelet leaders. As expected, we show that this method yields non-concave spectra and is not limited to their increasing part. From the theoretical point of view, we prove that this formalism is more efficient than the previous wavelet-based multifractal formalisms. We present the underlying function space and endow it with a topology.
Fonds de la Recherche Scientifique (Communauté française de Belgique) - F.R.S.-FNRS

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