Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Asymptotic regularity of Daubechies' scaling functions

Author(s): Ka-Sing Lau; Qiyu Sun
Journal: Proc. Amer. Math. Soc. 128 (2000), 1087-1095.
MSC (1991): Primary 42C15, 26A15, 26A18, 39A10, 42A05
Posted: July 28, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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.

References:

[1]
N. Bi, X. Dai and Q. Sun, Construction of compactly supported $M$-band wavelets, Appl. Comp. Harmonic Anal., to appear.

[2]
A. Cohen and I. Daubechies, Non-separable bidimensional wavelet bases, Rev. Mat. Iberoamericana, 9(1983), 51-137. MR 94k:42047

[3]
A. Cohen and I. Daubechies, A new method to determine the regularity of refinable functions, Rev. Mat. Iberoamericana, 12(1996), 527-591.

[4]
A. Cohen, I. Daubechies and A. Ron, How smooth is the smoothness function in a given refinable space?, Appl. Comput. Harmonic Anal., 3(1996), 87-89.

[5]
A. Cohen and E. Séré, Time-frequency localization with non-stationary wavelet packet, Preprint.

[6]
I. Daubechies, Ten Lectures on Wavelets, CBMS-Conference Lecture Notes, No. 61, SIAM Philadelphia, 1992. MR 93e:42045

[7]
I. Daubechies, Using Fredholm determinants to estimate the smoothness of refinable functions, In Approximation Theory VIII, Vol.2, Wavelet and Multilevel Approximation, Edited by C. K. Chui and L. L. Schumaker, World Scientific Press, 1995, pp. 89-122. MR 98e:42032

[8]
T. Eirola, Sobolev characterization of solutions of dilation equation, SIAM J. Math. Anal., 23(1992), 1015-1030. MR 93f:42056

[9]
A. Fan and K. S. Lau, Asymptotic behavior of multiperiodic periodic functions $G(x) = \prod _{n=1}^\infty g(x/2^n)$, J. Four. Anal. and Appl., 4(1998), 130-150.

[10]
L. Hervé, Construction et régularité des fonctions d'échelle, SIAM J. Math. Anal., 26(1995), 1361-1385. MR 97j:42015

[11]
K. S. Lau and J. Wang, Characterization of $L^p$-solution for the two-scale dilation equations, SIAM J. Math. Anal., 26(1995), 1018-1046. MR 96f:39004

[12]
B. Ma and Q. Sun, Compactly supported refinable distribution in Triebel-Lizorkin space and Besov space, J. Fourier Anal. Appl., to appear.

[13]
L. F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal., 25(1994), 1433-1466. MR 96f:39009

[14]
H. Volkmer, On the regularity of wavelets, IEEE Trans. Inform. Theory, 38(1992), 872-876. MR 93f:42058

[15]
H. Volkmer, Asymptotic regularity of compactly supported wavelets, SIAM J. Math. Anal., 26(1995), 1075-1087. MR 96h:42031

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42C15, 26A15, 26A18, 39A10, 42A05

Retrieve articles in all Journals with MSC (1991): 42C15, 26A15, 26A18, 39A10, 42A05

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 Mathematical 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

DOI: 10.1090/S0002-9939-99-05251-X
PII: S 0002-9939(99)05251-X
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
Posted: July 28, 1999
Communicated by: David R. Larson
Copyright of article: Copyright 2000, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google