Asymptotic regularity of Daubechies’ scaling functions

Abstract

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