Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )

A complete characterization is given of closed shift-invariant subspaces of L2(Rd) 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.

[1]  Klaus Höllig,et al.  B-splines from parallelepipeds , 1982 .

[2]  Ronald A. DeVore,et al.  Approximation by smooth multivariate splines , 1983 .

[3]  Klaus Höllig,et al.  Approximation order from bivariate ¹-cubics: a counterexample , 1983 .

[4]  Wolfgang Dahmen,et al.  Translates of multivarlate splines , 1983 .

[5]  C. Micchelli,et al.  On the approximation order from certain multivariate spline spaces , 1984, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[6]  Rong-Qing Jia,et al.  Controlled approximation and a characterization of the local approximation order , 1985 .

[7]  Rong-Qing Jia,et al.  Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh , 1986 .

[8]  Rong-Qing Jia.  A counterexample to a result concerning controlled approximation , 1986 .

[9]  Carl de Boor,et al.  The polynomials in the linear span of integer translates of a compactly supported function , 1987 .

[10]  I. J. Schoenberg.  Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions , 1988 .

[11]  I. R. H. Jackson.  An order of convergence for some radial basis functions , 1989 .

[12]  C. Micchelli,et al.  On multivariate -splines , 1989 .

[13]  A. Ron,et al.  Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems , 1990 .

[14]  M. Buhmann.  Multivariate cardinal interpolation with radial-basis functions , 1990 .

[15]  C. D. Boor,et al.  Quasiinterpolants and Approximation Power of Multivariate Splines , 1990 .

[16]  W. Madych,et al.  Polyharmonic cardinal splines , 1990 .

[17]  A. Ron.  A characterization of the approximation order for multivariate spline spaces , 1991 .

[18]  Martin D. Buhmann,et al.  Error Estimates for Multiquadric Interpolation , 1991, Curves and Surfaces.

[19]  R. Jia,et al.  Approximation by piecewise exponentials , 1991 .

[20]  Will Light,et al.  Recent Developments in the Strang-Fix Theory for Approximation Orders , 1991, Curves and Surfaces.

[21]  W. R. Madych.  Error Estimates for Interpolation by Generalized Splines , 1991, Curves and Surfaces.

[22]  Amos Ron,et al.  The Exponentials in the Span of the Multiinteger Translates of a Compactly Supported Function; Quasiinterpolation and Approximation Order , 1992 .

[23]  E. Cheney,et al.  Quasi-interpolation with translates of a function having noncompact support , 1992 .

[24]  A. Ron,et al.  On multivariate approximation by integer translates of a basis function , 1992 .

[25]  W. Light,et al.  On local and controlled approximation order , 1993 .

[26]  R. Jia,et al.  Approximation by multiinteger translates of functions having global support , 1993 .

[27]  N. Sivakumar,et al.  The approximation order of box spline spaces , 1993 .

[28]  M. Buhmann.  On quasi-interpolation with radial basis functions , 1993 .