Refinable subspaces of a refinable space
Authors:
Douglas P. Hardin and Thomas A. Hogan
Journal:
Proc. Amer. Math. Soc. 128 (2000), 19411950
MSC (1991):
Primary 39A10, 39B62, 42B99, 41A15
Published electronically:
October 29, 1999
MathSciNet review:
1662241
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Local refinable finitely generated shiftinvariant spaces play a significant role in many areas of approximation theory and geometric design. In this paper we present a new approach to the construction of such spaces. We begin with a refinable function which is supported on . We are interested in spaces generated by a function built from the shifts of .
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 C. de Boor, R. DeVore, Partitions of unity and approximation, Proc. Amer. Math. Soc. 93 (1985), 705709. MR 86f:41003
 [DM]
 W. Dahmen, C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293328. CMP 97:13
 [DL]
 I. Daubechies, J. C. Lagarias, Twoscale difference equations II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), 10311079. MR 93g:39001
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 T. N. T. Goodman, S. L. Lee, Refinable vectors of spline functions, Mathematical Methods for Curves and Surfaces II (M. Dæhlen, T. Lyche, L. L. Schumaker, eds.), Vanderbilt University Press, Nashville & London, 1997, pp. 213220. CMP 99:01
 [H]
 T. A. Hogan, A note on matrix refinement equations, SIAM J. Math. Anal. 29 (1998), 849854. CMP 98:11
 [HJ]
 T. A. Hogan, R.Q. Jia, Dependency relations among the shifts of a multivariate refinable distribution, Constr. Approx., to appear.
 [J1]
 R.Q. Jia, The Toeplitz theorem and its applications to approximation theory and linear PDE's, Trans. Amer. Math. Soc. 347 (1995), 25852594. MR 95i:41014
 [J2]
 R.Q. Jia, Shiftinvariant spaces on the real line, Proc. Amer. Math. Soc. 125 (1997), 785793. MR 97e:41039
 [J4]
 R.Q. Jia, Shiftinvariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259288. MR 99d:41016
 [J5]
 R.Q. Jia, Stability of the shifts of a finite number of functions, J. Approx. Theory 95 (1998), 194202. MR 99h:42040
 [J6]
 R.Q. Jia, Multiresolution of spaces, J. Math. Anal. Appl. 184 (1994), 620639. MR 95h:42034
 [JRZ]
 R.Q. Jia, S. D. Riemenschneider, and D. X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998), 15331563. MR 99d:42062
 [JS]
 Q. Jiang, Z. Shen, On existence and weak stability of matrix refinable functions, Constr. Approx., to appear.
 [LLS]
 W. Lawton, S. L. Lee, Z. Shen, Characterization of compactly supported refinable splines, Advances in Comp. Math. 3 (1995), 137145. MR 95m:41020
 [M]
 C. A. Micchelli, Mathematical Aspects of Geometric Modeling, CBMSNSF Regional Conference Series in Applied Mathematics v.65, SIAM, Philadelphia PA, 1995. MR 95i:65036
 [MP1]
 C. A. Micchelli, H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841870. MR 90k:65088
 [MP2]
 C. A. Micchelli, H. Prautzsch, Refinement and subdivision for spaces of integer translates of a compactly supported function, Numerical Analysis 1987 (D. F. Griffiths and G. A. Watson, eds.), Longman Scientific and Technical, Essex, 1987, pp. 192222. MR 90h:65016
 [MS]
 C. A. Micchelli, T. Sauer, Regularity of multiwavelets, Advances in Comp. Math. 7 (1997), 455545. CMP 98:01
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Additional Information
Douglas P. Hardin
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email:
hardin@math.vanderbilt.edu
Thomas A. Hogan
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email:
hogan@math.vanderbilt.edu
DOI:
http://dx.doi.org/10.1090/S0002993999052971
PII:
S 00029939(99)052971
Keywords:
Refinability,
matrix subdivision,
refinable function vector,
multiwavelet,
shiftinvariant,
FSI
Received by editor(s):
February 4, 1998
Received by editor(s) in revised form:
August 5, 1998
Published electronically:
October 29, 1999
Additional Notes:
This research was partially supported by a grant from the NSF and a grant from the Vanderbilt University Research Council.
Communicated by:
David R. Larson
Article copyright:
© Copyright 2000
American Mathematical Society
