Continuous wavelet transform of Schwartz tempered distributions

Lishan Liu,S.K Upadhyay,J.N Pandey

doi:10.1080/25742558.2019.1623647

Lishan Liu, S.K Upadhyay + Show 1 more

Open Access

https://doi.org/10.1080/25742558.2019.1623647

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Abstract

The continuous wavelet transform of Schwartz tempered distributions is investigated and derive the corresponding wavelet inversion formula (valid modulo a constant-tempered distribution) interpreti...

Highlights

Let SðRÞ be the Schwartz testing function space of rapid descent and let sðRÞ be a subspace of SðRÞ1ð so that every element φ 2 sðRÞ satisfies φðxÞdx 1⁄4 0, i.e., every element of sðRÞ is a basicÀ1 wavelet
One can verify that the restriction of f 2 S0ðRÞ to sðRÞ is in s0ðRÞ and, in the following discussion the wavelet inversion formula that is valid for f 2 S0ðRÞ restricted to SðRÞ modulo a constant distribution, is valid for elements of S0ðRÞ restricted to sðRÞ
We extend the continuous wavelet transform to the Schwartz tempered distribution space S0ðRÞ, exploiting the structure formula h f; φ i g; þ x2 Ámþ1 φðmþ1Þ ðxÞ

Summary

Introduction

1ð so that every element φ 2 sðRÞ satisfies φðxÞdx 1⁄4 0, i.e., every element of sðRÞ is a basic. Our objective is to extend the continuous wavelet transform to Schwartz space S0ðRÞ and prove an inversion formula modulo a constant distribution and extend the uniqueness theorem for the continuous wavelet transform of distributions to the space S0FðRÞ; the space SF0 ðRÞ is a subspace of the space S0ðRÞ. Ψ ÀxÀa bÁ; aÞ0; a; b 2 R a 1⁄4 0: Since any constant distribution in s0ðRÞ can be identified as a zero distribution, the uniqueness theorem for the wavelet inversion formula in s0ðRÞ is valid. (3.7) Theorem 3.6 [Inversion Formula]: Let f be a tempered distribution belonging to S0FðRÞ and ψðxÞ 2 sðRÞ & SðRÞ, and define Wf ða; bÞ of f with respect to the wavelet ψ by f ðtÞ; p1ffiffiffiffiffi ψ t À b jaj a. In view of (3.6) and Theorems 3.4 and 3.5, the integral in (3.11) is meaningful (it exists) and when operated against φ 2 SðRÞ, (3.11) becomes: hFðxÞ; φðxÞi

Cψ ð ð1 ð1

Conclusion