Regularity properties of random wavelet series

Esser, Céline; Jaffard, Stéphane; Vedel, Béatrice

doi:10.1090/tpms/1205

Authors: Céline Esser, Stéphane Jaffard and Béatrice Vedel
Journal: Theor. Probability and Math. Statist. 110 (2024), 31-53
MSC (2020): Primary 42C40; Secondary 42A20
DOI: https://doi.org/10.1090/tpms/1205
Published electronically: May 10, 2024
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the regularity properties of random wavelet series constructed by multiplying the coefficients of a deterministic wavelet series with unbounded I.I.D. random variables. In particular, we show that, at the opposite to what happens for Fourier series, the randomization of almost every continuous function gives an almost surely nowhere locally bounded function.

References

Alexandre Almeida, Wavelet bases in generalized Besov spaces, J. Math. Anal. Appl. 304 (2005), no. 1, 198–211. MR 2124658, DOI 10.1016/j.jmaa.2004.09.017
Jean-Marie Aubry and Stéphane Jaffard, Random wavelet series, Comm. Math. Phys. 227 (2002), no. 3, 483–514. MR 1910828, DOI 10.1007/s002200200630
Brian R. Hunt, Tim Sauer, and James A. Yorke, Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. MR 1161274, DOI 10.1090/S0273-0979-1992-00328-2
P. Billard, S. Kwapień, A. Pełczyński, and Ch. Samuel, Biorthogonal systems of random unconditional convergence in Banach spaces, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986) Longhorn Notes, Univ. Texas, Austin, TX, 1986, pp. 13–35. MR 1017039
Emile Borel, Sur les séries de Taylor, Acta Math. 21 (1897), no. 1, 243–247 (French). Lettre adressée à l’éditeur. MR 1554891, DOI 10.1007/BF02417979
Jens Peter Reus Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255–260 (1973). MR 326293, DOI 10.1007/BF02762799
Z. Ciesielski, Hölder conditions for realizations of Gaussian processes, Trans. Amer. Math. Soc. 99 (1961), 403–413. MR 132591, DOI 10.1090/S0002-9947-1961-0132591-2
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
Lara Daw and Laurent Loosveldt, Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process, Electron. J. Probab. 27 (2022), Paper No. 152, 45. MR 4515708, DOI 10.1214/22-ejp878
Karel de Leeuw, Yitzhak Katznelson, and Jean-Pierre Kahane, Sur les coefficients de Fourier des fonctions continues, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 16, A1001–A1003 (French, with English summary). MR 510870
Óscar Domínguez and Sergey Tikhonov, Function spaces of logarithmic smoothness: embeddings and characterizations, Mem. Amer. Math. Soc. 282 (2023), no. 1393, vii+166. MR 4539365, DOI 10.1090/memo/1393
David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard, Wavelet shrinkage: asymptopia?, J. Roy. Statist. Soc. Ser. B 57 (1995), no. 2, 301–369. With discussion and a reply by the authors. MR 1323344, DOI 10.1111/j.2517-6161.1995.tb02032.x
Céline Esser and Stéphane Jaffard, Divergence of wavelet series: a multifractal analysis, Adv. Math. 328 (2018), 928–958. MR 3771146, DOI 10.1016/j.aim.2018.02.010
Céline Esser and Laurent Loosveldt, Slow, ordinary and rapid points for Gaussian wavelets series and application to fractional Brownian motions, ALEA Lat. Am. J. Probab. Math. Stat. 19 (2022), no. 2, 1471–1495. MR 4517730, DOI 10.30757/alea.v19-59
A. Fraysse, Regularity criteria for almost every function in Sobolev spaces, J. Funct. Anal. 258 (2010), no. 6, 1806–1821. MR 2578456, DOI 10.1016/j.jfa.2009.11.017
Christopher Heil, A basis theory primer, Expanded edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2011. MR 2744776, DOI 10.1007/978-0-8176-4687-5
Stéphane Jaffard, Wavelet techniques for pointwise regularity, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 3–33 (English, with English and French summaries). MR 2225745, DOI 10.5802/afst.1111
Stéphane Jaffard and Yves Meyer, Wavelet methods for pointwise regularity and local oscillations of functions, Mem. Amer. Math. Soc. 123 (1996), no. 587, x+110. MR 1342019, DOI 10.1090/memo/0587
Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726, DOI 10.1017/CBO9780511526169
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
Jean-Pierre Kahane, A century of interplay between Taylor series, Fourier series and Brownian motion, Bull. London Math. Soc. 29 (1997), no. 3, 257–279. MR 1435557, DOI 10.1112/S0024609396002913
V. V. Kvaratskhelia, Unconditional convergence of functional series in problems of probability theory, J. Math. Sci. (N.Y.) 200 (2014), no. 2, 143–294. Translated from Sovrem. Mat. Prilozh. No. 86 (2013). MR 3306166, DOI 10.1007/s10958-014-1912-1
Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
Stéphane Mallat, A wavelet tour of signal processing, Academic Press, Inc., San Diego, CA, 1998. MR 1614527
M. B. Marcus and G. Pisier, Necessary and sufficient conditions for the uniform convergence of random trigonometric series, Lecture Notes Series, vol. 50, Aarhus Universitet, Matematisk Institut, Aarhus, 1978. Lecture 1977/78. MR 507061
Michael B. Marcus and Gilles Pisier, Random Fourier series with applications to harmonic analysis, Annals of Mathematics Studies, No. 101, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR 630532
Bernard Maurey, Au centre et autour de mathématiques de Jean-Pierre Kahane, SMF 2018: Congrès de la Société Mathématique de France, Sémin. Congr., vol. 33, Soc. Math. France, Paris, [2019] ©2019, pp. 1–67 (French, with French summary). MR 4508907
Yves Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR 1228209
F. L. Nazarov, The Bang solution of the coefficient problem, Algebra i Analiz 9 (1997), no. 2, 272–287 (Russian); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 407–419. MR 1468554
R. Paley and A. Zygmund, On some series of functions, Proc. Camb. Phil. Soc. 26 (1930), 266–272, 337–357, and 28 (1932) 458–474.
Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227. MR 116177, DOI 10.1512/iumj.1960.9.59013
Hugo Steinhaus, Über die Wahrscheinlichkeit dafür, daßder Konvergenzkreis einer Potenzreihe ihre natürliche Grenze ist, Math. Z. 31 (1930), no. 1, 408–416 (German). MR 1545123, DOI 10.1007/BF01246422
Jan-Olov Strömberg, A modified Franklin system and higher-order spline systems on $\textbf {R}^{n}$ as unconditional bases for Hardy spaces, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 475–494. MR 730086
Norbert Wiener, Differential-space, J. Math. and Phys. 2 (1923), 131–174. MR 4250033, DOI 10.1002/sapm192321131
Yves Meyer, Fabrice Sellan, and Murad S. Taqqu, Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion, J. Fourier Anal. Appl. 5 (1999), no. 5, 465–494. MR 1755100, DOI 10.1007/BF01261639