On Wavelet and Leader Wavelet Based Large Deviation Multifractal Formalisms for Non-uniform Hölder Functions

• Published in: The Journal of fourier analysis and applications 2019-04, Vol.25 (2), p.506-522
• Main Authors: Ben Slimane, Mourad, Ben Abid, Moez, Ben Omrane, Ines, Halouani, Borhen
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Deviation; Formalism; Fourier Analysis; Function space; Histograms; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Signal,Image and Speech Processing; Toruses; Visibility; Wavelet analysis
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-017-9578-y

Recently, a large deviation multifractal formalism based on histograms of wavelet leader coefficients, compared to some other wavelet-based formalisms, was proved to be efficient for uniform Hölder functions. In this paper, we extend this efficiency for non-uniform Hölder functions. We first obtain optimal bounds for both wavelet and wavelet leader histograms for all functions in the critical Besov space B t m / t , q ( T ) , where t , q > 0 and T is the unit torus of R m . We then compute these histograms for quasi-all functions in B t m / t , q ( T ) , in the sense of Baire Category. Although, increasing parts of these histograms have increasing visibility, they coincide only if 0 < q ≤ t . If moreover q ≤ 1 , then wavelet leader histograms method covers the Hölder spectrum for all t > 0 , however wavelet histograms method covers it only if 0 < q ≤ t .