A class of -dilation scaling functions

with regularity growing proportionally

to filter support width

Authors:
Xianliang Shi and Qiyu Sun

Journal:
Proc. Amer. Math. Soc. **126** (1998), 3501-3506

MSC (1991):
Primary 42C15

DOI:
https://doi.org/10.1090/S0002-9939-98-05070-9

MathSciNet review:
1626478

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, a class of -dilation scaling functions with regularity growing proportionally to filter support width is constructed. This answers a question proposed by Daubechies on p.338 of her book *Ten Lectures on Wavelets* (1992).

**[C]**C. K. Chui,*An Introduction to Wavelets*, Academic Press, 1992. MR**93f:42055****[CL]**C. K. Chui and J. Lian, Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale ,*Appl. Comput. Harmonic Anal.*, 2(1995), 21-51. MR**95m:42042****[BDS]**N. Bi, X. Dai and Q. Sun, Construction of compactly supported -band wavelets,*Appl. Comput. Harmonic Anal.*, To appear.**[D]**I. Daubechies,*Ten Lectures on Wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics 61, Philadephia, 1992. MR**93e:42045****[DL]**I. Daubechies and J. Lagarias, Two-scale difference equation I: existence and global regularity of solution,*SIAM J. Math. Anal*., 22(1991), 1388-1410. MR**92d:39001****[H]**P. N. Heller, Rank wavelets with vanishing moments,*SIAM J. Matrix Anal. Appl*., 16(1995), 502-519. MR**95k:42058****[HW1]**P. N. Heller and R. O. Wells, Jr., The spectral theory of multiresolution operators and applications, In*Wavelets: Theory, Algorithms, and Applications*, edited by C. K. Chui, L. Montefusco, and L. Puccio, Academic Press, 1994, pp. 13-42. MR**96a:42046****[HW2]**P. N. Heller ad R. O. Wells Jr., Sobolev regularity for rank wavelets, CML Technical Report TR 96-08, Computational Mathematics Laboratory, Rice University, 1996.**[HSZ]**D. Huang, Q. Sun and Z. Zhang, Integral representation of -band filter of Daubechies type,*Chinese Sciences Bulletin*, 42(1997), 803-807. CMP**97:17****[M]**Y. Meyer,*Ondelettes et Opérateurs,I: Ondelettes*, Hermann, Paris, 1990. MR**93i:42002****[So]**P. M. Soardi, Hölder regularity of compactly supported -wavelets,*Constr. Approx.*, To appear.**[S]**Q. Sun, Sobolev exponent estimates and asymptotic regularity of band Daubechies' scaling functions, Preprint, 1997.**[WL]**G. V. Welland and M. Lundberg, Construction of compact -wavelets,*Constr. Approx.*, 9(1993), 347-370. MR**94i:42049**

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Additional Information

**Xianliang Shi**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Email:
xshi@math.tamu.edu

**Qiyu Sun**

Affiliation:
Center for Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China

Address at time of publication:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

Email:
matsunqy@leonis.nus.edu.sg

DOI:
https://doi.org/10.1090/S0002-9939-98-05070-9

Received by editor(s):
November 20, 1995

Additional Notes:
The first author is supported by the Texas Higher Education Coordinating Board under Grant Number 999903-109. The second author is partially supported by the National Natural Sciences Foundation of China # 69735020, the Tian Yuan Foundation, the Doctoral Bases Promotion Foundation of National Educational Commission of China # 97033519 and the Zhejiang Provincial Sciences Foundation of China # 196083, and by the Wavelets Strategic Research Program, National University of Singapore, under a grant from the National Science and Technology Board and the Ministry of Education, Singapore.

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 1998
American Mathematical Society