Divergence of wavelet series: A multifractal analysis

Esser, Céline; Jaffard, Stéphane

doi:10.1016/j.aim.2018.02.010

Download

Article (Scientific journals)

Divergence of wavelet series: A multifractal analysis

Esser, Céline; Jaffard, Stéphane

2018 • In Advances in Mathematics, 328, p. 928-958

Peer Reviewed verified by ORBi

Permalink
https://hdl.handle.net/2268/205478

DOI
10.1016/j.aim.2018.02.010

Files (1)Send to Details Statistics Bibliography Similar publications

Files

Full Text

1701.02982v1.pdf

Author preprint (449.81 kB)

Download

All documents in ORBi are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Baire genericity; Besov sequence spaces; Divergence properties; Hausdorff dimension; Lineability; Multifractal Analysis; Prevalence; Wavelet Series

Abstract :

[en] We show the relevance of a multifractal-type analysis for pointwise convergence and divergence properties of wavelet series: Depending on the sequence space which the wavelet coefficients sequence belongs to, we obtain deterministic upper bounds for the Hausdorff dimensions of the sets of points where a given rate of divergence occurs, and we show that these bounds are generically optimal, according to several notions of genericity.

Disciplines :

Mathematics

Author, co-author :

Esser, Céline ; Université de Liège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Jaffard, Stéphane ; Université de Liège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Language :

English

Title :

Divergence of wavelet series: A multifractal analysis

Publication date :

2018

Journal title :

Advances in Mathematics

ISSN :

0001-8708

eISSN :

1090-2082

Publisher :

Elsevier, Atlanta, United States

Volume :

328

Pages :

928-958

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 17 January 2017

Statistics

Number of views

69 (7 by ULiège)

Number of downloads

100 (6 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Aubry, J.-M., On the rate of pointwise divergence of Fourier and wavelet series in Lp. J. Approx. Theory 538 (2006), 97–111.
Aubry, J.-M., Maman, D., Seuret, S., Local behavior of traces of Besov functions: prevalent results. J. Funct. Anal. 264:3 (2013), 631–660.
Ayache, A., Esser, C., A useful result related with zeros of continuous compactly supported mother wavelets. Int. J. Wavelets Multiresolut. Inf. Process., 15(5), 2017, 1750044 12 p.
Barral, J., Durand, A., Jaffard, S., Seuret, S., Local multifractal analysis. Carfi, D., Lapidus, M.L., Pearse, E.P.J., van Frankenhuijsen, M., (eds.) Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics Contemp. Math., vol. 601, 2013, Amer. Math. Soc., Providence, 31–64.
Bayart, F., Heurteaux, Y., Multifractal analysis of the divergence of Fourier series. Ann. Sci. Éc. Norm. Supér. 45 (2012), 927–946.
Bayart, F., Heurteaux, Y., Multifractal analysis of the divergence of Fourier series: the extreme cases. J. Anal. Math. 124 (2014), 387–408.
Bayart, F., Heurteaux, Y., Multifractal phenomena and packing dimension. Rev. Mat. Iberoam. (to appear) arXiv:1610.00981.
Benedetto, J.J., Li, S., The theory of multiresolution analysis and applications to filter banks. Appl. Comput. Harmon. Anal., 5, 1998.
Bernal-González, L., Algebraic genericity of strict order integrability. Studia Math. 199:3 (2010), 279–293.
Bernal-González, L., Pellegrino, D., Seoane-Sepúlveda, J.B., Linear subsets of nonlinear sets in topological vector spaces. Bull. Amer. Math. Soc. 51:1 (2014), 71–130.
Christensen, J.P.R., Topology and Borel Structure. 1974, North-Holland, Amsterdam.
Cohen, A., Daubechies, I., Feauveau, J.-C., Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45 (1992), 485–560.
Daubechies, I., Ten Lectures on Wavelets. 1992, SIAM.
Daubechies, I., Grossmann, A., Meyer, Y., Painless nonorthogonal expansions. J. Math. Phys. 27 (1996), 1271–1283.
Fraysse, A., Regularity criteria of almost every function in a Sobolev space. J. Funct. Anal. 258 (July 2010), 1806–1821.
Fraysse, A., Jaffard, S., How smooth is almost every function in a Sobolev space?. Rev. Mat. Iberoam. 22 (2006), 663–682.
Hunt, B.R., Sauer, T., Yorke, J.A., Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.) 27:2 (1992), 217–238.
Jaffard, S., Wavelets and nonlinear analysis. Benedetto, J., Frazier, M., (eds.) Wavelets: Mathematics and Applications Stud. Adv. Math., 1994, CRC, 313–329.
Jaffard, S., On the Frisch–Parisi conjecture. J. Math. Pures Appl. 79:6 (2000), 525–552.
Kelly, S.E., Kon, M.A., Raphael, L.A., Local convergence for wavelet expansion. J. Funct. Anal. 126 (1994), 102–138.
Kelly, S.E., Kon, M.A., Raphael, L.A., Pointwise convergence of wavelet expansion. Bull. Amer. Math. Soc. 30 (1994), 87–94.
Kostadinova, S., Vindas, J., Multiresolution expansions of distributions: pointwise convergence and quasiasymptotic behavior. Acta Appl. Math. 138 (2015), 115–134.
Meise, R., Vogt, D., Introduction to Functional Analysis. 1997, Clarendon Press.
Meyer, Y., Ondelettes et opérateurs. 1990, Hermann.
Pietsch, A., Eigenvalues of integral operators. I. Math. Ann. 247 (1980), 169–178.
Tao, T., On the almost everywhere convergence of wavelet summation methods. Appl. Comput. Harmon. Anal., 3(4), 1996.
Triebel, H., Theory of Function Spaces II. 1992, Birkhauser.
Walter, G.G., Pointwise convergence of wavelet expansions. J. Approx. Theory 80 (1995), 108–118.
Zayed, A.I., Pointwise convergence of a class of non-orthogonal wavelet expansions. Proc. Amer. Math. Soc. 128:12 (2000), 3629–3637.