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Function series with multifractal variations
Author(s) -
Barral Julien,
Seuret Stephane
Publication year - 2004
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410199
Subject(s) - multifractal system , series (stratigraphy) , mathematics , classification of discontinuities , measure (data warehouse) , jump , function (biology) , piecewise linear function , piecewise , mathematical analysis , wavelet , spectrum (functional analysis) , statistical physics , fractal , physics , quantum mechanics , paleontology , database , evolutionary biology , artificial intelligence , computer science , biology
In this paper, we study three classes of multifractal function series using a work achieved for a new class of measures defined in [3]. The originality of these function series consists in the fact that the sizes of the jumps (or of the amplitudes of the pulses) depend on the location of the jump points and on a measure μ . In particular, there may be a strong heterogeneity in the distribution of the size of the jumps. These function series f are defined by$$ f(x) = \sum \limits _{j \ge 1} {1 \over {j^2}} \sum \limits ^{b^{j} -1} _{b=0} \mu ( [ kb^{-j} , (k+1) b^{-j} ] ) \psi_{j, k} (x) \, , $$where ψ j,k is a contracted and dilated version of a single function ψ . This function ψ will either be a wavelet, a pulse, or a piecewise linear function. We show that under suitable conditions on the measure μ , the multifractal spectrum of f can be computed. For large classes of measures, the spectrum is linear between 0 and a critical value h c , and if h ≥ h c , f and μ share the same spectrum. This untypical shape is the result of the combination of the multifractal measure μ with the rapid variations or discontinuities of the functions ψ j,k . (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)