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2019 Continuous wavelet transform of schwartz distributions
J.N. Pandey
Rocky Mountain J. Math. 49(6): 2005-2028 (2019). DOI: 10.1216/RMJ-2019-49-6-2005

Abstract

In this paper we extend the continuous wavelet transform to Schwartz distributions and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. But the uniqueness theorem for our wavelet inversion formula is valid for the space $\cal D '_F$ obtained by filtering (deleting) (i) all non-zero constant distributions from the space $\cal D '$, (ii) all non-zero constants that appear with a distribution as a union, as for example, for $ {x^2}/(1+x^2) = 1-1/(1+x^2)$, 1 is deleted and $-1/(1+x^2)$ is retained. The kernel of our wavelet transform is an element of $\cal D $ which when integrated along the real line vanishes, but none of its moments of order $m\ge 1$ along the real line is zero. The set of such kernels will be denoted by $D_m$.

Citation

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J.N. Pandey. "Continuous wavelet transform of schwartz distributions." Rocky Mountain J. Math. 49 (6) 2005 - 2028, 2019. https://doi.org/10.1216/RMJ-2019-49-6-2005

Information

Published: 2019
First available in Project Euclid: 3 November 2019

zbMATH: 07136591
MathSciNet: MR4027246
Digital Object Identifier: 10.1216/RMJ-2019-49-6-2005

Subjects:
Primary: ‎42C40, 46F12
Secondary: 46F05, 46F10

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 6 • 2019
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