Rational Orthonormal Littlewood-Paley Wavelet Basis of L2(R)

TL;DR

This paper identifies limitations in existing rational Littlewood-Paley wavelet bases and introduces a new orthonormal basis that works for all rational dilation factors, proving its validity for L2(R).

Contribution

It proposes a new rational Littlewood-Paley wavelet basis valid for all rational dilation factors, overcoming previous restrictions.

Findings

The existing basis is not orthonormal for non-integer rational dilation factors.

A new basis is constructed that is orthonormal for all rational numbers.

The new basis is proven to be an orthonormal wavelet basis of L2(R).

Abstract

In this letter, first, we prove that the orthonormal basis of rational Littlewood-Paley wavelet with rational dilation factor M=p/q first proposed by Auscher does not hold for all rational numbers. It does not hold if q is not equal to 1. In other words, it is not an orthonormal basis if the rational dilation factor M is not an integer. Then, to make up for the shortcoming of the rational Littlewood-Paley wavelet proposed by Auscher, a new orthonormal basis of rational Littlewood-Paley wavelet with rational dilation factor M=p/q is proposed, which holds for all rational numbers. Finally, by means of sampling theorem for bandpass signals, it is proved completely that the new rational Littlewood-Paley wavelet family is an orthonormal wavelet basis of L2(R).