Approximation from locally finite-dimensional
shift-invariant spaces
Author:
Kang Zhao
Journal:
Proc. Amer. Math. Soc. 124 (1996), 1857-1867
MSC (1991):
Primary 41A15, 41A25, 41A28, 41A63
DOI:
https://doi.org/10.1090/S0002-9939-96-03253-4
MathSciNet review:
1307577
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: After exploring some topological properties of locally finite-dimensional shift-invariant subspaces of
, we show that if
provides approximation order
, then it provides the corresponding simultaneous approximation order. In the case
is generated by a compactly supported function in
, it is proved that
provides approximation order
in the
-norm with
if and only if the generator is a derivative of a compactly supported function that satisfies the Strang-Fix conditions.
- 1. Carl de Boor, Quasiinterpolants and approximation power of multivariate splines, Computation of curves and surfaces (Puerto de la Cruz, 1989) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 307, Kluwer Acad. Publ., Dordrecht, 1990, pp. 313–345. MR 1064965
- 2. Carl de Boor, Approximation order without quasi-interpolants, Approximation theory VII (Austin, TX, 1992) Academic Press, Boston, MA, 1993, pp. 1–18. MR 1212567
- 3. C. de Boor and R. DeVore, Partitions of unity and approximation, Proc. Amer. Math. Soc. 93 (1985), no. 4, 705–709. MR 776207, https://doi.org/10.1090/S0002-9939-1985-0776207-2
- 4. 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, https://doi.org/10.1090/S0002-9947-1994-1195508-X
- 5. C. de Boor and G. J. Fix, Spline approximation by quasiinterpolants, J. Approximation Theory 8 (1973), 19–45. MR 340893, https://doi.org/10.1016/0021-9045(73)90029-4
- 6. W. A. Light and E. W. Cheney, Quasi-interpolation with translates of a function having noncompact support, Constr. Approx. 8 (1992), no. 1, 35–48. MR 1142692, https://doi.org/10.1007/BF01208904
- 7. Rong Qing Jia, A characterization of the approximation order of translation invariant spaces of functions, Proc. Amer. Math. Soc. 111 (1991), no. 1, 61–70. MR 1010801, https://doi.org/10.1090/S0002-9939-1991-1010801-1
- 8. ------, The Topelitz theorem and its applications to approximation theory and linear PDE's, Proc. Amer. Math. Soc. (to appear). CMP 94:13
- 9. Rong Qing Jia and Junjiang Lei, Approximation by multi-integer translates of functions having global support, J. Approx. Theory 72 (1993), no. 1, 2–23. MR 1198369, https://doi.org/10.1006/jath.1993.1002
- 10. Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
- 11. I. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Parts A & B, Quart. Appl. Math. 4 (1946), 45--99, 112--141. MR 8:55d
- 12. G. Strang and G. Fix, A Fourier analysis of the finite element variation method, C.I.M.E.II (Ciclo 1971) (G. Geymonat, ed.), Constructive Aspects of Functional Analysis, 1973, pp. 793--840.
- 13. K. Zhao, Simultaneous approximation from PSI spaces, J. Approx. Theory (to appear).
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Additional Information
Kang Zhao
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Address at time of publication:
Structural Dynamics Research Corporation, 2000 Eastman Dr., Milford, Ohio 45150
Email:
kang.zhao@sdrc.com
DOI:
https://doi.org/10.1090/S0002-9939-96-03253-4
Keywords:
Approximation order,
locally finite-dimensional,
polynomial reproducing,
shift-invariant space,
simultaneous approximation,
Strang-Fix condition
Received by editor(s):
June 28, 1994
Received by editor(s) in revised form:
December 13, 1994
Communicated by:
J. Marshall Ash
Article copyright:
© Copyright 1996
American Mathematical Society