Harmonic analysis
Regularity properties of Haar Frames
Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1107-1117.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.228
Classification: 42B35, 42C40, 46E35, 65T60, 68T05
Jaffard, Stéphane 1; Krim, Hamid 2

1 Univ Paris Est Creteil, CNRS, LAMA, F-94010 Creteil, France, Univ Gustave Eiffel, LAMA, F-77447 Marne-la-Vallée, France.
2 Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695 USA.
@article{CRMATH_2021__359_9_1107_0,
     author = {Jaffard, St\'ephane and Krim, Hamid},
     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},
     url = {http://www.numdam.org/articles/10.5802/crmath.228/}
}
TY  - JOUR
AU  - Jaffard, Stéphane
AU  - Krim, Hamid
TI  - Regularity properties of Haar Frames
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1107
EP  - 1117
VL  - 359
IS  - 9
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.228/
DO  - 10.5802/crmath.228
LA  - en
ID  - CRMATH_2021__359_9_1107_0
ER  - 
%0 Journal Article
%A Jaffard, Stéphane
%A Krim, Hamid
%T Regularity properties of Haar Frames
%J Comptes Rendus. Mathématique
%D 2021
%P 1107-1117
%V 359
%N 9
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.228/
%R 10.5802/crmath.228
%G en
%F CRMATH_2021__359_9_1107_0
Jaffard, Stéphane; Krim, Hamid. Regularity properties of Haar Frames. Comptes Rendus. Mathématique, Volume 359 (2021) no. 9, pp. 1107-1117. doi : 10.5802/crmath.228. http://www.numdam.org/articles/10.5802/crmath.228/

[1] Battle, Guy A block spin construction of ondelettes. Part II: The QFT connection, Commun. Math. Phys., Volume 114 (1988), pp. 93-102 | DOI | MR

[2] Bourdaud, Gérard Ondelettes et espaces de Besov, Rev. Mat. Iberoam., Volume 11 (1995) no. 3, pp. 477-512 | DOI | MR | Zbl

[3] Bruna, Joan; Zaremba, Wojciech; Szlam, Arthur; Lecun, Yann 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 (Bengio, Yoshua; LeCun, Yann, eds.), Volume 27, ICLR (2016)

[4] Cheng, Xiuyuan; Chen, Xu; Mallat, Stéphane Deep Haar scattering networks, Inf. Inference, Volume 5 (2016), pp. 105-133 | DOI | MR | Zbl

[5] Christensen, Ole An introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, 2003 | DOI | Zbl

[6] Coifman, Ronald R.; Donoho, David L. 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 (Antoniadis Anestis, Oppenheim G. et al., eds.) (Lecture Notes in Statistics), Volume 103, Springer, 1995, pp. 121-150 | Zbl

[7] Daubechies, Ingrid; Grossmann, Alex; Meyer, Yves Painless nonorthogonal expansions, J. Math. Phys., Volume 27 (1986) no. 5, pp. 1271-1283 | DOI | MR | Zbl

[8] Daubechies, Ingrid; Lagarias, Jeffrey C. On the Thermodynamic Formalism for Multifractal Functions, Rev. Math. Phys., Volume 6 (1994), pp. 1033-1070 | DOI | MR | Zbl

[9] DeVore, Ronald A.; Lorentz, George B. Constructive approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer, 1993 | MR | Zbl

[10] DeVore, Ronald A.; Richards, Franklin Saturation and inverse theorems for spline approximation, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) (Meir, A.; Sharma, A., eds.) (International Series of Numerical Mathematics), Volume 21, Birkhäuser, 1973, pp. 73-82 | MR | Zbl

[11] Heil, Christopher; Jorgensen, Palle E. T.; Larson, David R. 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] Jaffard, Stéphane 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] Jaffard, Stéphane Wavelet techniques in multifractal analysis, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Multifractals, probability and statistical mechanics, applications (Lapidus, Michel; van Frankenhuijsen, M., eds.) (Proceedings of Symposia in Pure Mathematics), Volume 72(2), American Mathematical Society (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] Jaffard, Stéphane; Mandelbrot, Benoît B. 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] Jaffard, Stéphane; Martin, Bruno Multifractal analysis of the Brjuno function, Invent. Math., Volume 212 (2018) no. 1, pp. 109-132 | DOI | MR | Zbl

[16] Krim, Hamid; Jaffard, Stéphane; Roheda, S.; Mahdizadehaghdam, Shahin; Panahi, A. On Stabilizing Generative Adversarial Networks (STGANS) (2021) (in preparation)

[17] LeCun, Yann; Bengio, Yoshua; Hinton, Goeffrey Deep learning, Nature, Volume 521 (2015), pp. 436-444 | DOI

[18] Lemarié, Piere-Gilles Ondelettes à localisation exponentielle, J. Math. Pures Appl., Volume 67 (1988) no. 3, pp. 227-236 | MR | Zbl

[19] Lemarié, Piere-Gilles; Meyer, Yves F. Ondelettes et bases hilbertiennes, Rev. Mat. Iberoam., Volume 2 (1986) no. 1-2, pp. 1-18 | DOI | MR | Zbl

[20] Li, Ming; Ma, Zheng; Wang, Yu Guang; Zhuang, Xiaosheng Fast Haar Transform for Graph Neural Networks, Neural Networks, Volume 128 (2020), pp. 188-198 | DOI | Zbl

[21] Meyer, Yves Ondelettes et Opérateurs. I : Ondelettes. II : Opérateurs de Caldéron–Zygmund, Actualités Mathématiques, Hermann, 1990 | Zbl

[22] Pesquet, Jean-Christophe; Krim, Hamid; Carfantan, Hervé Time-invariant orthonormal wavelet representations, IEEE Trans. Signal Process., Volume 44 (1996) no. 8, pp. 1964-1970 | DOI

[23] Pesquet, Jean-Christophe; Krim, Hamid; Carfantan, Hervé; Proakis, John G. Estimation of noisy signals using time-invariant wavelet packets, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, Volume 1, IEEE (1993), pp. 31-34 | DOI

[24] Tao, Terence On the almost everywhere convergence of wavelet summation methods, Appl. Comput. Harmon. Anal., Volume 3 (1996) no. 4, pp. 384-387 | MR | Zbl

[25] Walter, Gilbert G. Pointwise convergence of wavelet expansions, J. Approx. Theory, Volume 80 (1995) no. 1, pp. 108-118 | DOI | MR | Zbl

Cited by Sources: