Properties and applications of transversal operators

TL;DR

This paper explores properties of transversal operators, specifically translation and dilation, demonstrating their unitarity and usefulness in various analysis systems like Fourier and wavelet analysis.

Contribution

It introduces and analyzes the properties of transversal operators T and D, highlighting their applications in multiple analysis frameworks.

Findings

Transversal operators T and D are unitary.

These operators are useful in Fourier, Gabor, and wavelet analysis.

Demonstrated their role in proofs within analytic systems.

Abstract

This paper presents some properties and applications of "transversal operators". Two transversal operators are presented: a "translation" operator T and a "dilation" operator D. Such operators are used in common analysis systems including Fourier series analysis, Fourier analysis, Gabor analysis, multiresolution analysis (MRA), and wavelet analysis. Like the unitary Fourier transform operator F, the transversal operators T and D are unitary. Demonstrations of the usefulness of these three unitary operators are found in the proofs of results found in some common analytic systems including MRA analysis and wavelet analysis.