Multidimensional Fractional Wavelet Transforms and Uncertainty Principles

TL;DR

This paper introduces a new multidimensional fractional wavelet transform, explores its properties, and establishes uncertainty principles analogous to classical Fourier analysis, expanding the theoretical framework of wavelet transforms.

Contribution

It defines the multidimensional fractional wavelet transform (MFrWT), studies its fundamental properties, and derives uncertainty principles for it based on those of the multidimensional fractional Fourier transform.

Findings

MFrWT range is a reproducing kernel Hilbert space

Derived Heisenberg, logarithmic, and local uncertainty principles for MFrWT

Established connections between MFrWT and MFrFT uncertainty principles

Abstract

In this paper, we have given a new definition of continuous fractional wavelet transform in $\mathbb{R}^N$, namely the multidimensional fractional wavelet transform (MFrWT) and studied some of the basic properties along with the inner product relation and the reconstruction formula. We have also shown that the range of the proposed transform is a reproducing kernel Hilbert space and obtain the associated kernel. We have obtained the uncertainty principle like Heisenberg's uncertainty principle, logarithmic uncertainty principle and local uncertainty principle of the multidimensional fractional Fourier transform (MFrFT). Based on these uncertainty principles of the MFrFT we have obtained the corresponding uncertainty principles i.e., Heisenberg's, logarithmic and local uncertainty principles for the proposed MFrWT.