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arXiv · 2511.08984

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

Changping Sun
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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).

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Changping Sun. 2025-11-12. Rational Orthonormal Littlewood-Paley Wavelet Basis of L2(R). https://arxiv.org/abs/2511.08984

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