Rational Orthonormal Littlewood-Paley 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).
Cite
@article{arxiv.2511.08984,
title = {Rational Orthonormal Littlewood-Paley Wavelet Basis of L2(R)},
author = {Changping Sun},
journal= {arXiv preprint arXiv:2511.08984},
year = {2025}
}