Asymptotic regularity of Daubechies’ scaling functions
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- by Ka-Sing Lau and Qiyu Sun PDF
- Proc. Amer. Math. Soc. 128 (2000), 1087-1095 Request permission
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.References
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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
- MR Author ID: 190087
- 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
- Email: matsunqy@leonis.nus.edu.sg
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1087-1095
- MSC (1991): Primary 42C15, 26A15, 26A18, 39A10, 42A05
- DOI: https://doi.org/10.1090/S0002-9939-99-05251-X
- MathSciNet review: 1654093