Elliptic-cylindrical Wavelets: The Mathieu Wavelets

TL;DR

This paper introduces Mathieu wavelets, a new family of wavelets based on Mathieu differential equations, with potential applications in optics and electromagnetism, offering customizable filter notches.

Contribution

It presents a novel wavelet family derived from Mathieu equations, establishing a multiresolution analysis linked to Floquet's solutions, and enabling easy design of filter notches.

Findings

Wavelets are based on Mathieu differential equations.

Filters have customizable notch characteristics.

Potential applications in optics and electromagnetism.

Abstract

This note introduces a new family of wavelets and a multiresolution analysis, which exploits the relationship between analysing filters and Floquet's solution of Mathieu differential equations. The transfer function of both the detail and the smoothing filter is related to the solution of a Mathieu equation of odd characteristic exponent. The number of notches of these filters can be easily designed. Wavelets derived by this method have potential application in the fields of Optics and Electromagnetism.