Year: 2026
Type: preprint
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 $12$ for $12$ 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.
Source: arXiv (Cornell University)
Author: Nielsen, Morten
Institution: Aalborg University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214711771