On a Conjecture about Schauder-Basis Properties of the Daubechies Wavelet Packets

Nielsen and Zhou (\emph{Mean size of wavelet packets}, ACHA 13 (2002), 22–34) conjectured that for every Daubechies filter of length at least four the associated wavelet packets fail to be a Schauder basis of $L^p(\mathbb{R})$ for every $p\mathrm{\ne }2$; their own ${\mathrm{\ell }}^1\mathrm{/}{\mathrm{\ell }}^{\mathrm{\infty }}$ estimate only reaches extreme exponents. We prove the Schauder-basis-failure half of the conjecture in full, for every $1<p<\mathrm{\infty }$ with $p\mathrm{\ne }2$, in the length-four case. The proof combines convexity of the wavelet packet pressure function with the uniqueness of equilibrium states for irreducible matrix families due to Feng and Käenmäki (2011) and an exact algebraic separation of two periodic spectral growth rates of the high-pass transition matrices.