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Refinable subspaces of a refinable space


Authors: Douglas P. Hardin and Thomas A. Hogan
Journal: Proc. Amer. Math. Soc. 128 (2000), 1941-1950
MSC (1991): Primary 39A10, 39B62, 42B99, 41A15
DOI: https://doi.org/10.1090/S0002-9939-99-05297-1
Published electronically: October 29, 1999
MathSciNet review: 1662241
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Abstract: Local refinable finitely generated shift-invariant 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 $\psi :\mathbb{R}\to \mathbb{R}^{m}$ which is supported on $[0,1]$. We are interested in spaces generated by a function $\phi :\mathbb{R}\to \mathbb{R}^{n}$ built from the shifts of $\psi $.


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  • [BD] C. de Boor, R. DeVore, Partitions of unity and approximation, Proc. Amer. Math. Soc. 93 (1985), 705-709. MR 86f:41003
  • [DM] W. Dahmen, C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293-328. CMP 97:13
  • [DL] I. Daubechies, J. C. Lagarias, Two-scale difference equations II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), 1031-1079. MR 93g:39001
  • [GL] 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. 213-220. CMP 99:01
  • [H] T. A. Hogan, A note on matrix refinement equations, SIAM J. Math. Anal. 29 (1998), 849-854. 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), 2585-2594. MR 95i:41014
  • [J2] R.-Q. Jia, Shift-invariant spaces on the real line, Proc. Amer. Math. Soc. 125 (1997), 785-793. MR 97e:41039
  • [J4] R.-Q. Jia, Shift-invariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259-288. MR 99d:41016
  • [J5] R.-Q. Jia, Stability of the shifts of a finite number of functions, J. Approx. Theory 95 (1998), 194-202. MR 99h:42040
  • [J6] R.-Q. Jia, Multiresolution of $L_{p}$ spaces, J. Math. Anal. Appl. 184 (1994), 620-639. MR 95h:42034
  • [JRZ] R.-Q. Jia, S. D. Riemenschneider, and D. X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998), 1533-1563. 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), 137-145. MR 95m:41020
  • [M] C. A. Micchelli, Mathematical Aspects of Geometric Modeling, CBMS-NSF 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), 841-870. 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. 192-222. MR 90h:65016
  • [MS] C. A. Micchelli, T. Sauer, Regularity of multiwavelets, Advances in Comp. Math. 7 (1997), 455-545. 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: https://doi.org/10.1090/S0002-9939-99-05297-1
Keywords: Refinability, matrix subdivision, refinable function vector, multiwavelet, shift-invariant, 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

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