The limits of refinable functions

Authors:
Gilbert Strang and Ding-Xuan Zhou

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1971-1984

MSC (2000):
Primary 42C40, 41A25; Secondary 65F15

Published electronically:
January 4, 2001

MathSciNet review:
1813602

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

A function is refinable ( ) if it is in the closed span of . This set is not closed in , and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.

**[1]**Carl de Boor, Ronald A. DeVore, and Amos Ron,*Approximation from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})*, Trans. Amer. Math. Soc.**341**(1994), no. 2, 787–806. MR**1195508**, 10.1090/S0002-9947-1994-1195508-X**[2]**J. O. Chapa,*Matched wavelet construction and its application to target detection*, Ph.D. thesis, Rochester Institute of Technology, 1995.**[3]**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****[4]**Ingrid Daubechies and Jeffrey C. Lagarias,*Two-scale difference equations. I. Existence and global regularity of solutions*, SIAM J. Math. Anal.**22**(1991), no. 5, 1388–1410. MR**1112515**, 10.1137/0522089**[5]**Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust,*Fractal functions and wavelet expansions based on several scaling functions*, J. Approx. Theory**78**(1994), no. 3, 373–401. MR**1292968**, 10.1006/jath.1994.1085**[6]**Christopher Heil, Gilbert Strang, and Vasily Strela,*Approximation by translates of refinable functions*, Numer. Math.**73**(1996), no. 1, 75–94. MR**1379281**, 10.1007/s002110050185**[7]**Henry Helson,*Lectures on invariant subspaces*, Academic Press, New York-London, 1964. MR**0171178****[8]**Rong-Qing Jia,*Shift-invariant spaces on the real line*, Proc. Amer. Math. Soc.**125**(1997), no. 3, 785–793. MR**1350950**, 10.1090/S0002-9939-97-03586-7**[9]**Rong Qing Jia and Charles A. Micchelli,*Using the refinement equations for the construction of pre-wavelets. II. Powers of two*, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 209–246. MR**1123739****[10]**R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou,*Approximation by multiple refinable functions*, Canad. J. Math.**49**(1997), no. 5, 944–962. MR**1604122**, 10.4153/CJM-1997-049-8**[11]**Rong-Qing Jia, S. D. Riemenschneider, and Ding-Xuan Zhou,*Vector subdivision schemes and multiple wavelets*, Math. Comp.**67**(1998), no. 224, 1533–1563. MR**1484900**, 10.1090/S0025-5718-98-00985-5**[12]**R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou,*Smoothness of multiple refinable functions and multiple wavelets*, SIAM J. Matrix Anal. Appl.**21**(1999), 1-28. CMP**2000:01****[13]**Yves Meyer,*Wavelets and operators*, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992. Translated from the 1990 French original by D. H. Salinger. MR**1228209****[14]**Gilbert Strang and Truong Nguyen,*Wavelets and filter banks*, Wellesley-Cambridge Press, Wellesley, MA, 1996. MR**1411910****[15]**G. Strang and V. Strela,*Orthogonal multiwavelets with vanishing moments*, Optical Eng.**33**(1994), 2104-2107.**[16]**Gilbert Strang and Ding-Xuan Zhou,*Inhomogeneous refinement equations*, J. Fourier Anal. Appl.**4**(1998), no. 6, 733–747. MR**1666013**, 10.1007/BF02479677**[17]**Ding Xuan Zhou,*Construction of real-valued wavelets by symmetry*, J. Approx. Theory**81**(1995), no. 3, 323–331. MR**1333755**, 10.1006/jath.1995.1054**[18]**Ding-Xuan Zhou,*Existence of multiple refinable distributions*, Michigan Math. J.**44**(1997), no. 2, 317–329. MR**1460417**, 10.1307/mmj/1029005707

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Additional Information

**Gilbert Strang**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
gs@math.mit.edu

**Ding-Xuan Zhou**

Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kow- loon, Hong Kong, P. R. China

Email:
mazhou@math.cityu.edu.hk

DOI:
https://doi.org/10.1090/S0002-9947-01-02668-X

Keywords:
Refinable function,
Fourier transform,
band-limited function,
refinement mask,
inhomogeneous refinement equation,
multiple refinable function,
fully refinable function

Received by editor(s):
May 15, 1998

Received by editor(s) in revised form:
November 3, 1999

Published electronically:
January 4, 2001

Additional Notes:
Research supported in part by Research Grants Council of Hong Kong.

Article copyright:
© Copyright 2001
American Mathematical Society