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arXiv · 2609.32904

Mean size and Schauder-basis properties of Daubechies wavelet packets

Morten Nielsen
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Abstract

In 2002 Nielsen and Zhou conjectured that the shifted wavelet packet system associated with a Daubechies filter of length at least four fails to be a Schauder basis of $L^p(\mathbb{R})$ for $1\le p\le\infty$, $p\ne2$, and that the packets are not uniformly bounded in $p$-mean across scales for any $p>2$. The basis obstruction throughout the range $1 2$ for $1 2$ the full family of transition matrices has $p$-norm joint spectral radius exceeding the Haar value $4^{1/p}$. The key step is structural: an affine piece of the matrix pressure forces constant spectral radius, which by a theorem of Protasov and Voynov forces simultaneous orthogonality of the transition matrices, incompatible with the double zero of the high-pass symbol. The results extend to every real spectral factor with the Daubechies magnitude response, including the least-asymmetric filters.

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Morten Nielsen. 2026-09-26. Mean size and Schauder-basis properties of Daubechies wavelet packets. https://arxiv.org/abs/2609.32904

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