Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Continuous Wavelet Transform of Schwartz Tempered Distributions in $S'(\mathbb R^n)$

Version 1 : Received: 11 January 2019 / Approved: 14 January 2019 / Online: 14 January 2019 (10:08:33 CET)

A peer-reviewed article of this Preprint also exists.

Pandey, J.N.; Maurya, J.S.; Upadhyay, S.K.; Srivastava, H.M. Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n ). Symmetry 2019, 11, 235. Pandey, J.N.; Maurya, J.S.; Upadhyay, S.K.; Srivastava, H.M. Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n ). Symmetry 2019, 11, 235.

Journal reference: Symmetry 2019, 11, 235
DOI: 10.3390/sym11020235

## Abstract

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

## Subject Areas

function spaces and their duals; distributions; generalized functions; distribution space; wavelet transform of generalized functions

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