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).