Asymptotic regularity of Daubechies’ scaling functions

Abstract:

Let $\phi _N$, $N\ge 1$, be Daubechies’ scaling function with symbol $\big ({1+e^{-i\xi }\over 2}\big )^N Q_N(\xi )$, and let $s_p(\phi _N),0<p\le \infty$, be the corresponding $L^p$ Sobolev exponent. In this paper, we make a sharp estimation of $s_p(\phi _N)$, and we prove that there exists a constant $C$ independent of $N$ such that \[ N-{\ln |Q_N(2\pi /3)|\over \ln 2}-{C\over N}\le s_p(\phi _N)\le N-{\ln |Q_N(2\pi /3)|\over \ln 2}. \] This answers a question of Cohen and Daubeschies (Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.