Daubechies小波包的均值大小与Schauder基性质
Mean size and Schauder-basis properties of Daubechies wavelet packets
- Aalborg University（奥尔堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了对长度至少为4的Daubechies滤波器，其平移小波包系统在$L^p$中不是Schauder基（$p\ne2$），且$p$-均值无界，通过矩阵压力结构迫使谱半径常数并导致正交性矛盾，并扩展到实谱因子。
AI中文摘要：
2002年，Nielsen和Zhou猜想：与长度至少为4的Daubechies滤波器相关联的平移小波包系统，对于$1\le p\le\infty$且$p\ne2$，不是$L^p(\mathbb{R})$的Schauder基，并且对于任何$p>2$，这些小波包在尺度上的$p$-均值不是一致有界的。对于长度4的情况，通过显式计算，此前已在$1<p<\infty$且$p\ne2$的整个范围内建立了基的障碍；完整的$p$-均值断言仍然未解决。我们证明了对每个长度至少为4的Daubechies滤波器，这两个猜想都成立：对于$1<p<\infty$且$p\ne2$，高通转移矩阵满足$\rho_p\rho_{p'}>2$；对于$p>2$，整个转移矩阵族的$p$-范数联合谱半径超过Haar值$4^{1/p}$。关键步骤是结构性的：矩阵压力的一个仿射片段迫使谱半径为常数，根据Protasov和Voynov的一个定理，这迫使转移矩阵同时正交，这与高通符号的双零点不相容。结果扩展到具有Daubechies幅度响应的每个实谱因子，包括最小非对称滤波器。
英文摘要：
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 $1<p<\infty$, $p\ne2$, 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 $1<p<\infty$, $p\ne2$, 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.
