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Asymptotic regularity
of Daubechies' scaling functions

Authors: Ka-Sing Lau and Qiyu Sun
Journal: Proc. Amer. Math. Soc. 128 (2000), 1087-1095
MSC (1991): Primary 42C15, 26A15, 26A18, 39A10, 42A05
Published electronically: July 28, 1999
MathSciNet review: 1654093
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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

\begin{displaymath}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}. \end{displaymath}

This answers a question of Cohen and Daubeschies ( Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.

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Additional Information

Ka-Sing Lau
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260; Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, Hong Kong

Qiyu Sun
Affiliation: Center for Mthematical Sciences, Zhejiang University, Hangzhou 310027, China
Address at time of publication: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore

Keywords: Fourier transform, scaling function, Sobolev exponent, wavelet
Received by editor(s): November 3, 1997
Received by editor(s) in revised form: May 30, 1998
Published electronically: July 28, 1999
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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