Mean size and Schauder-basis properties of Daubechies wavelet packets
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 Lp(ℝ) for 1≤ p≤∞, p≠2, and that the packets are not uniformly bounded in p-mean across scales for any p>2. The basis obstruction throughout the range 1<p<∞, p≠2, was previously established for length four, via an explicit computation; the full p-mean assertion remained open. We prove both for every Daubechies filter of length at least four: the high-pass transition matrices satisfy ρpρp'>2 for 1<p<∞, p≠2, and for p>2 the full family of transition matrices has p-norm joint spectral radius exceeding the Haar value 41/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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BibTeX
@misc{nielsen2026mean,
title = {{Mean size and Schauder-basis properties of Daubechies wavelet packets}},
author = {Morten Nielsen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32904},
eprint = {2609.32904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32904},
}
RIS
TY - UNPB
TI - Mean size and Schauder-basis properties of Daubechies wavelet packets
AU - Morten Nielsen
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32904
UR - https://arxiv.org/abs/2609.32904
AB - 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(ℝ) for 1≤ p≤∞, p≠2, and that the packets are not uniformly bounded in p-mean across scales for any p>2. The basis obstruction throughout the range 1<p<∞, p≠2, was previously established for length four, via an explicit computation; the full p-mean assertion remained open. We prove both for every Daubechies filter of length at least four: the high-pass transition matrices satisfy ρ_pρ_p'>2 for 1<p<∞, p≠2, and for p>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.
ER -