Linear canonical wavelet transform and the associated uncertainty principles

TL;DR

This paper introduces the linear canonical wavelet transform (LCWT), explores its mathematical properties, and establishes uncertainty principles and bounds related to its time-frequency analysis capabilities.

Contribution

It presents the novel LCWT, analyzes its key properties, and derives uncertainty principles specific to this new transform.

Findings

Established inner product and reconstruction formulas for LCWT.

Proved Donoho-Stark's and Lieb's uncertainty principles for LCWT.

Provided a lower bound for the measure of LCWT's essential support.

Abstract

We define a novel time-frequency analyzing tool, namely linear canonical wavelet transform (LCWT) and study some of its important properties like inner product relation, reconstruction formula and also characterize its range. We obtain Donoho-Stark's and Lieb's uncertainty principle for the LCWT and give a lower bound for the measure of its essential support. We also give Shapiro's mean dispersion theorem for the proposed LCWT.