Refinable subspaces of a refinable space

Hardin, Douglas; Hogan, Thomas

doi:10.1090/S0002-9939-99-05297-1

Refinable subspaces of a refinable space
HTML articles powered by AMS MathViewer

by Douglas P. Hardin and Thomas A. Hogan PDF

Proc. Amer. Math. Soc. 128 (2000), 1941-1950 Request permission

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 $\psi$.

References

C. de Boor and R. DeVore, Partitions of unity and approximation, Proc. Amer. Math. Soc. 93 (1985), no. 4, 705–709. MR 776207, DOI 10.1090/S0002-9939-1985-0776207-2
W. Dahmen, C. A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997), 293–328.
Ingrid Daubechies and Jeffrey C. Lagarias, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), no. 4, 1031–1079. MR 1166574, DOI 10.1137/0523059
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.
T. A. Hogan, A note on matrix refinement equations, SIAM J. Math. Anal. 29 (1998), 849–854.
T. A. Hogan, R.-Q. Jia, Dependency relations among the shifts of a multivariate refinable distribution, Constr. Approx., to appear.
Rong Qing Jia, The Toeplitz theorem and its applications to approximation theory and linear PDEs, Trans. Amer. Math. Soc. 347 (1995), no. 7, 2585–2594. MR 1277117, DOI 10.1090/S0002-9947-1995-1277117-8
Rong-Qing Jia, Shift-invariant spaces on the real line, Proc. Amer. Math. Soc. 125 (1997), no. 3, 785–793. MR 1350950, DOI 10.1090/S0002-9939-97-03586-7
Rong-Qing Jia, Shift-invariant spaces and linear operator equations, Israel J. Math. 103 (1998), 259–288. MR 1613580, DOI 10.1007/BF02762276
Rong-Qing Jia, Stability of the shifts of a finite number of functions, J. Approx. Theory 95 (1998), no. 2, 194–202. MR 1652860, DOI 10.1006/jath.1998.3215
Rong Qing Jia, Multiresolution of $L_p$ spaces, J. Math. Anal. Appl. 184 (1994), no. 3, 620–639. MR 1281533, DOI 10.1006/jmaa.1994.1226
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, DOI 10.1090/S0025-5718-98-00985-5
Q. Jiang, Z. Shen, On existence and weak stability of matrix refinable functions, Constr. Approx., to appear.
Wayne Lawton, S. L. Lee, and Zuowei Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995), no. 1-2, 137–145. MR 1314906, DOI 10.1007/BF03028364
Charles A. Micchelli, Mathematical aspects of geometric modeling, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 65, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR 1308048, DOI 10.1137/1.9781611970067
Charles A. Micchelli and Hartmut Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841–870. MR 986909, DOI 10.1016/0024-3795(89)90495-3
C. A. Micchelli and H. Prautzsch, Refinement and subdivision for spaces of integer translates of a compactly supported function, Numerical analysis 1987 (Dundee, 1987) Pitman Res. Notes Math. Ser., vol. 170, Longman Sci. Tech., Harlow, 1988, pp. 192–222. MR 951929
C. A. Micchelli, T. Sauer, Regularity of multiwavelets, Advances in Comp. Math. 7 (1997), 455-545.