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Approximation from shift-invariant subspaces of $ L\sb 2(\bold R\sp d)$


Authors: Carl de Boor, Ronald A. DeVore and Amos Ron
Journal: Trans. Amer. Math. Soc. 341 (1994), 787-806
MSC: Primary 41A25; Secondary 41A63, 42B10, 46E20
DOI: https://doi.org/10.1090/S0002-9947-1994-1195508-X
MathSciNet review: 1195508
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Abstract: A complete characterization is given of closed shift-invariant subspaces of $ {L_2}({\mathbb{R}^d})$ which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.


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  • [A] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [B1] C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), 199-208. MR 889555 (88e:41054)
  • [B2] -, Quasiinterpolants and approximation power of multivariate splines, Computation of Curves and Surfaces (M. Gasca and C. A. Micchelli, eds.), Kluwer Academic Publishers, Dordrecht, Netherlands, 1990, pp. 313-345. MR 1064965 (91i:41009)
  • [Bu1] M. D. Buhmann, Multivariate interpolation with radial basis functions, Constr. Approx. 6 (1990), 225-256. MR 1054754 (91f:41001)
  • [Bu2] -, On quasi-interpolation with radial basis functions, J. Approx. Theory 72 (1993), 103-130. MR 1198376 (94a:41038)
  • [BD] C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), 775-788. MR 688977 (84j:41015)
  • [BuD] M. D. Buhmann and N. Dyn, Error estimates for multiquadric interpolation, Curves and Surfaces (P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds.), Academic Press, New York, 1990, pp. 1-4. MR 1123718
  • [BH1] C. de Boor and K. Höllig, $ B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/1983), 99-115. MR 729403 (86d:41008)
  • [BH2] -, Approximation order from bivariate $ {C^1}$-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), 649-655. MR 687634 (84j:41014)
  • [BJ] C. de Boor and R. Q. Jia, Controlled approximation and a characterization of the local approximation order, Proc. Amer. Math. Soc. 95 (1985), 547-553. MR 810161 (87d:41025)
  • [BR1] C. de Boor and A. Ron, The exponentials in the span of the integer translates of a compactly supported function: approximation orders and quasi-interpolation, J. London Math. Soc. 45 (1992), 519-535. MR 1180260 (94e:41021)
  • [BR2] -, Fourier analysis of approximation orders from principal shift-invariant spaces, Constr. Approx. 8 (1992), 427-462. MR 1194028 (94c:41023)
  • [C] C. K. Chui, Multivariate splines, CBMS-NSF Regional Conference Ser. Appl. Math., no. 54, SIAM, Philadelphia, Pa., 1988. MR 1033490 (92e:41009)
  • [CL] E. W. Cheney and W. A. Light, Quasi-interpolation with basis functions having non-compact support, Constr. Approx. 8 (1992), 35-48. MR 1142692 (93a:41027)
  • [DJLR] N. Dyn, I. R. H. Jackson, D. Levin, and A. Ron, On multivariate approximation by the integer translates of a basis function, Israel J. Math. 78 (1992), 95-130. MR 1194962 (94i:41024)
  • [DM1] W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/3 (1983), 217-234. MR 709352 (85e:41033)
  • [DM2] -, On the approximation order from certain multivariate spline spaces, J. Austral. Math. Soc. Ser. B 26 (1984), 233-246. MR 765640 (87j:41032)
  • [DR] N. Dyn and A. Ron, Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems, Trans. Amer. Math. Soc. 319 (1990), 381-404. MR 956032 (90i:41020)
  • [H] H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964. MR 0171178 (30:1409)
  • [HL] E. J. Halton and W. A. Light, On local and controlled approximation order, J. Approx. Theory 72 (1993), 268-277. MR 1209967 (94i:41038)
  • [J1] R. Q. Jia, Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh, Trans. Amer. Math. Soc. 295 (1986), 199-212. MR 831196 (88d:41018)
  • [J2] -, A counterexample to a result concerning controlled approximation, Proc. Amer. Math. Soc. 97 (1986), 647-654. MR 845982 (87j:41037)
  • [JL] R. Q. Jia and J. J. Lei, Approximation by multiinteger translates of functions having global support, J. Approx. Theory 72 (1993), 2-23. MR 1198369 (94f:41024)
  • [Ja] I. R. H. Jackson, An order of convergence for some radial basis functions, IMA J. Numer. Anal. 9 (1989), 567-587. MR 1030648 (91e:65019)
  • [L] W. A. Light, Recent developments in the Strang-Fix theory for approximation orders, Curves and Surfaces (P. J. Laurent, A. le Méhauté, and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 285-292. MR 1123748
  • [LJ] J. J. Lei and R. Q. Jia, Approximation by piecewise exponentials, SIAM J. Math. Anal. 22 (1991), 1776-1789. MR 1129411 (92i:41016)
  • [M] W. R. Madych, Error estimates for interpolation by generalized splines, Curves and Surfaces (P. J. Laurent, A. le Méhauté, and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 297-306. MR 1123750
  • [MN1] W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines. I, J. Approx. Theory 40 (1990), 141-156. MR 1033167 (90j:41022)
  • [MN2] -, Multivariate interpolation and conditionally positive functions. II, Math. Comp. 54 (1990), 211-230. MR 993931 (90e:41007)
  • [P] M. J. D. Powell, The theory of radial basis function approximation in 1990, Advances in Numerical Analysis, Vol. II: Wavelets, Subdivision Algorithms and Radial Basis Functions (W. A. Light, ed.), Oxford Univ. Press, 1992, pp. 105-210. MR 1172121
  • [R] A. Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1) (1991), 73-90. MR 1110099 (92g:41017)
  • [Ra] C. Rabut, Polyharmonic cardinal $ B$-splines, Parts A and B, preprint, 1989.
  • [RS] A. Ron and N. Sivakumar, The approximation order of box spline spaces, Proc. Amer. Math. Soc. 117 (1993), 473-482. MR 1110553 (93d:41010)
  • [S] I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Parts A and B, Quart. Appl. Math. 4 (1946), 45-99; 112-141.
  • [SF] G. Strang and G. Fix, A Fourier analysis of the finite element variational method, C.I.M.E. II, Ciclo 1971, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), 1973, pp. 793-840.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1195508-X
Keywords: Approximation order, Strang-Fix conditions, shift-invariant spaces, radial basis functions, orthogonal projection
Article copyright: © Copyright 1994 American Mathematical Society

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