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 | References | Similar Articles | Additional Information

Abstract: Let , , be Daubechies' scaling function with symbol , and let , be the corresponding Sobolev exponent. In this paper, we make a sharp estimation of , and we prove that there exists a constant independent of such that

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

**[1]**N. Bi, X. Dai and Q. Sun, Construction of compactly supported -band wavelets,*Appl. Comp. Harmonic Anal.*, to appear.**[2]**Albert Cohen and Ingrid Daubechies,*Nonseparable bidimensional wavelet bases*, Rev. Mat. Iberoamericana**9**(1993), no. 1, 51–137. MR**1216125**, 10.4171/RMI/133**[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]**Ingrid Daubechies,*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107****[7]**Ingrid Daubechies,*Using Fredholm determinants to estimate the smoothness of refinable functions*, Approximation theory VIII, Vol. 2 (College Station, TX, 1995) Ser. Approx. Decompos., vol. 6, World Sci. Publ., River Edge, NJ, 1995, pp. 89–112. MR**1471776****[8]**Timo Eirola,*Sobolev characterization of solutions of dilation equations*, SIAM J. Math. Anal.**23**(1992), no. 4, 1015–1030. MR**1166573**, 10.1137/0523058**[9]**A. Fan and K. S. Lau, Asymptotic behavior of multiperiodic periodic functions ,*J. Four. Anal. and Appl.*, 4(1998), 130-150.**[10]**Loïc Hervé,*Construction et régularité des fonctions d’échelle*, SIAM J. Math. Anal.**26**(1995), no. 5, 1361–1385 (French, with English and French summaries). MR**1347425**, 10.1137/S0036141092240023**[11]**Ka-Sing Lau and Jianrong Wang,*Characterization of 𝐿^{𝑝}-solutions for the two-scale dilation equations*, SIAM J. Math. Anal.**26**(1995), no. 4, 1018–1046. MR**1338372**, 10.1137/S0036141092238771**[12]**B. Ma and Q. Sun, Compactly supported refinable distribution in Triebel-Lizorkin space and Besov space,*J. Fourier Anal. Appl.*, to appear.**[13]**Lars F. Villemoes,*Wavelet analysis of refinement equations*, SIAM J. Math. Anal.**25**(1994), no. 5, 1433–1460. MR**1289147**, 10.1137/S0036141092228179**[14]**Hans Volkmer,*On the regularity of wavelets*, IEEE Trans. Inform. Theory**38**(1992), no. 2, 872–876. MR**1162224**, 10.1109/18.119743**[15]**Hans Volkmer,*Asymptotic regularity of compactly supported wavelets*, SIAM J. Math. Anal.**26**(1995), no. 4, 1075–1087. MR**1338375**, 10.1137/S0036141093248967

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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

Email:
matsunqy@leonis.nus.edu.sg

DOI:
https://doi.org/10.1090/S0002-9939-99-05251-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

Published electronically:
July 28, 1999

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society