A note on continuous fractional wavelet transform in $\mathbb{R}^n$

TL;DR

This paper explores the properties and mathematical foundations of the continuous fractional wavelet transform in multi-dimensional space, including admissibility, reconstruction, uncertainty principles, and boundedness on Morrey spaces.

Contribution

It establishes new conditions for admissibility, derives key relations, and analyzes boundedness of the CFrWT in higher dimensions, extending existing wavelet theory.

Findings

Derived admissibility conditions using fractional convolution.

Established inner product, reconstruction, and reproducing kernel formulas.

Proved Heisenberg's and local uncertainty inequalities for the transform.

Abstract

In this paper, we have studied continuous fractional wavelet transform (CFrWT) in $n$-dimensional Euclidean space $\mathbb{R}^n$ with dilation parameter $\boldsymbol a=(a_{1},a_{2},\ldots,a_{n}),$ such that none of $a_{i}'s$ are zero. Necessary and sufficient condition for the admissibility of a function is established with the help of fractional convolution. Inner product relation, reconstruction formula and the reproducing kernel for the CFrWT depending on two wavelets are obtained. Heisenberg's uncertainty inequality and Local uncertainty inequality for the CFrWT are obtained. Finally, boundedness of the transform on the Morrey space $L^{1,\nu}_{M}(\mathbb{R}^n)$ and the estimate of $L^{1,\nu}_{M}(\mathbb{R}^n)$-distance of the CFrWT of two argument functions with respect to different wavelets are discussed.