On multivariate approximation by integer translates of a basis function
 N. Dyn,
 I. R. H. Jackson,
 D. Levin,
 A. Ron
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Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasiinterpolation is addressed first. The analysis makes an essential use of the structure of the generalized Fourier transform of the basis function. In contrast with spline theory, polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasiinterpolants. These convergence orders are established by a careful and quite involved examination of the decay rates of the basis function. Furthermore, it is shown that the same approximation orders are obtained with quasiinterpolants defined on a bounded domain.
 Title
 On multivariate approximation by integer translates of a basis function
 Journal

Israel Journal of Mathematics
Volume 78, Issue 1 , pp 95130
 Cover Date
 199202
 DOI
 10.1007/BF02801574
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 N. Dyn ^{(1)}
 I. R. H. Jackson ^{(2)}
 D. Levin ^{(3)}
 A. Ron ^{(4)}
 Author Affiliations

 1. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
 2. Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, England
 3. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
 4. Computer Sciences Department, University of WisconsinMadison, 53706, Madison, WI, USA