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

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.