Abstract
Spherically symmetric continuous wavelet decompositions are considered, and the notion of Riesz means is introduced for them. Generalized localization is proved for the decompositions under study in Lp classes without any restrictions on the wavelets. Further, generalized localization is studied for the Riesz means of wavelet decompositions of distributions from the Sobolev class with negative order of smoothness.
Similar content being viewed by others
References
V. A. Il’in, “The generalized treatment of the principle of localization for Fourier series in fundamental systems of functions,” Sibirsk. Mat. Zh. 9 (5), 1093–1106 (1968).
A. Carbery and F. Soria, “Almost everywhere convergence of Fourier integrals forfunctions in Sobolev spaces, and L2 localization principle,” Rev. Mat. Iberoamericana 4 (2), 319–337 (1988).
A. Carbery and F Soria, “Pointwise Fourier inversion and localization in Rn,” J. Fourier Anal. Appl. 3 (Special Issue) (1997).
A. Bastis, “Generalized localization of Fourier series with respect to the eigenfunctions of the Laplace operator in the classes L p,” Litovsk. Mat. Sb. 31 (3), 387–405 (1991).
R. Ashurov and A. Butaev, “On generalized localization of Fourier inversion for distributions,” in Topics in Functional Analysis and Algebra, Contemp. Math. (Amer. Math. Soc, Providence, RI, 2016). Vol. 672, pp. 33–50.
R. Ashurov, A. Butaev, and B. Pradhan, “On generalized localization of Fourier inversion associated with an elliptic operatorfor distributions,” Abstr. Appl. Anal., No. Art. ID 649848 (2012).
Sh. A. Alimov, “Generalized localization of Riesz means of spectral expansions of distributions,” Dokl. Ross. Akad.Nauk 446 (1) 7–9 (2012) [Dokl. Math. 86 (2), 597-599 (2012)].
R. Ashurov and A. Butaev, “On pointwise convergence of continuous wavelet transforms,” Uzbek Math. J., No. 1 4–26 (2018).
I. Daubechies, Ten Lectures on Wavelets (Philadelphia, PA, SIAM, 1992).
M. Rao, H. Šikić, and R. Song, “Application of Carleson’s theorem to wavelet inversion,” Control Cybernet. 23 (4) 761–771 (1994).
R. Ashurov and A. Butaev, “On continuous wavelet transforms of distributions,” Appl. Math. Lett. 24 (9), 1578–1583 (2011).
R. Ashurov and A. Butaev, “On spherically symmetric continuous wavelet transforms of functions from Liouville classes,” Int. J. Math. Comput. 11 (111), 111–117 (2011).
Acknowledgments
The author wishes to express gratitude to Sh. A. Alimov for discussions of the results.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 6, pp. 803-810.
Rights and permissions
About this article
Cite this article
Ashurov, R.R., Faiziev, Y.É. Generalized Localization Principle for Continuous Wavelet Decompositions. Math Notes 106, 857–863 (2019). https://doi.org/10.1134/S0001434619110208
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434619110208