Inversion formula and Parsval theorem for complex continuous wavelet transforms studied by entangled state representation

TL;DR

This paper develops the inversion formula and Parseval theorem for complex continuous wavelet transforms using entangled state representation, establishing a complete theoretical framework and revealing new properties of mother wavelets.

Contribution

It introduces the inversion formula and Parseval theorem for CCWT via entangled state representation, completing the theoretical foundation.

Findings

Derived the inversion formula for CCWT.

Established Parseval theorem for CCWT.

Discovered a new orthogonal property of mother wavelet in parameter space.

Abstract

In a preceding Letter (Opt. Lett. 32, 554 (2007)) we have proposed complex continuous wavelet transforms (CCWTs) and found Laguerre--Gaussian mother wavelets family. In this work we present the inversion formula and Parsval theorem for CCWT by virtue of the entangled state representation, which makes the CCWT theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.