We prove that pointwise and global Hölder regularity can be characterized using the coefficients on the Haar tight frame obtained by using a finite union of shifted Haar bases, despite the fact that the elements composing the frame are discontinuous.
Révisé le :
Accepté le :
Publié le :
CC-BY 4.0
@article{CRMATH_2021__359_9_1107_0,
author = {St\'ephane Jaffard and Hamid Krim},
title = {Regularity properties of {Haar} {Frames}},
journal = {Comptes Rendus. Math\'ematique},
pages = {1107--1117},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {9},
year = {2021},
doi = {10.5802/crmath.228},
language = {en},
}
Stéphane Jaffard; Hamid Krim. Regularity properties of Haar Frames. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1107-1117. doi : 10.5802/crmath.228. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.228/
[1] A block spin construction of ondelettes. Part II: The QFT connection, Commun. Math. Phys., Volume 114 (1988), pp. 93-102 | DOI | MR
[2] Ondelettes et espaces de Besov, Rev. Mat. Iberoam., Volume 11 (1995) no. 3, pp. 477-512 | DOI | MR | Zbl
[3] Spectral networks and locally connected networks on graphs, 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, Volume 27 (2016)
[4] Deep Haar scattering networks, Inf. Inference, Volume 5 (2016), pp. 105-133 | DOI | MR | Zbl
[5] An introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, 2003 | DOI | Zbl
[6] Translation-Invariant de-noising, Wavelets and Statistics. Proceedings of the 15th French-Belgian meeting of statisticians, held at Villard de Lans, France, November 16-18, 1994 (Oppenheim G. Antoniadis Anestis, ed.) (Lecture Notes in Statistics), Volume 103, Springer, 1995, pp. 121-150 | Zbl
[7] Painless nonorthogonal expansions, J. Math. Phys., Volume 27 (1986) no. 5, pp. 1271-1283 | DOI | MR | Zbl
[8] On the Thermodynamic Formalism for Multifractal Functions, Rev. Math. Phys., Volume 6 (1994), pp. 1033-1070 | DOI | MR | Zbl
[9] Constructive approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer, 1993 | MR | Zbl
[10] Saturation and inverse theorems for spline approximation, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) (A. Meir; A. Sharma, eds.) (International Series of Numerical Mathematics), Volume 21, Birkhäuser, 1973, pp. 73-82 | MR | Zbl
[11] Wavelets, Frames, and Operator Theory. Papers from the Focused Research Group Workshop, University of Maryland, College Park, MD, USA, January 15–21, 2003, Contemporary Mathematics, 345, American Mathematical Society, 2004 | Zbl
[12] Exposants de Hölder en des points donnés et coefficients d’ondelettes, C. R. Math. Acad. Sci. Paris, Volume 308 (1989) no. 4, pp. 79-81 | Zbl
[13] Wavelet techniques in multifractal analysis, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Multifractals, probability and statistical mechanics, applications (Proceedings of Symposia in Pure Mathematics), Volume 72(2) (2004), pp. 91-152 (in part the proceedings of a special session held during the annual meeting of the American Mathematical Society, San Diego, CA, USA, January 2002) | Zbl
[14] Local regularity of nonsmooth wavelet expansions and application to Polya’s function, Adv. Math., Volume 120 (1996) no. 2, pp. 265-282 | DOI | MR | Zbl
[15] Multifractal analysis of the Brjuno function, Invent. Math., Volume 212 (2018) no. 1, pp. 109-132 | DOI | MR | Zbl
[16] On Stabilizing Generative Adversarial Networks (STGANS) (2021) (in preparation)
[17] Deep learning, Nature, Volume 521 (2015), pp. 436-444 | DOI
[18] Ondelettes à localisation exponentielle, J. Math. Pures Appl., Volume 67 (1988) no. 3, pp. 227-236 | MR | Zbl
[19] Ondelettes et bases hilbertiennes, Rev. Mat. Iberoam., Volume 2 (1986) no. 1-2, pp. 1-18 | DOI | MR | Zbl
[20] Fast Haar Transform for Graph Neural Networks, Neural Networks, Volume 128 (2020), pp. 188-198 | DOI | Zbl
[21] Ondelettes et Opérateurs. I : Ondelettes. II : Opérateurs de Caldéron–Zygmund, Actualités Mathématiques, Hermann, 1990 | Zbl
[22] Time-invariant orthonormal wavelet representations, IEEE Trans. Signal Process., Volume 44 (1996) no. 8, pp. 1964-1970 | DOI
[23] Estimation of noisy signals using time-invariant wavelet packets, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, Volume 1 (1993), pp. 31-34 | DOI
[24] On the almost everywhere convergence of wavelet summation methods, Appl. Comput. Harmon. Anal., Volume 3 (1996) no. 4, pp. 384-387 | MR | Zbl
[25] Pointwise convergence of wavelet expansions, J. Approx. Theory, Volume 80 (1995) no. 1, pp. 108-118 | DOI | MR | Zbl
Cité par Sources :
Commentaires - Politique