Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Controlled approximation and a characterization of the local approximation order

Authors: C. de Boor and R.-Q. Jia
Journal: Proc. Amer. Math. Soc. 95 (1985), 547-553
MSC: Primary 41A25; Secondary 65N30
MathSciNet review: 810161
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The local approximation order from a scale $ ({S_h})$ of approximating functions on $ {{\mathbf{R}}^m}$ is characterized in terms of the linear span (and its Fourier transform) of the finitely many compactly supported functions $ \varphi $ whose integer translates $ \varphi ( \cdot - j),j \in {z^m}$, span the space $ S = {S_1}$ from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix.

References [Enhancements On Off] (What's this?)

  • [BH] C. de Boor and K. Höllig, $ B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99-115. MR 729403 (86d:41008)
  • [DM] W. Dahmen and C. A. Micchelli, On the approximation order from certain multivariate spline spaces, J. Austral. Math. Soc. Ser B 26 (1984), 233-246. MR 765640 (87j:41032)
  • [FS] G. Fix and G. Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory, Stud. Appl. Math. 48 (1969), 265-273. MR 0258297 (41:2944)
  • [J] R.-q. Jia, A counterexample to a result of Strang and Fix concerning controlled approximation, MRC TSR# 2743, 1984.
  • [R] W. Rudin, Function theory in the unit ball of $ {C^n}$, Grundlehren Math. Wiss., Vol. 241, Springer-Verlag, New York, 1980. MR 601594 (82i:32002)
  • [Sc] I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, A, B, Quart. Appl. Math. 4 (1946), 45-99, 112-141.
  • [St] G. Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations. II, SYNSPADE 1970 (B. Hubbard, ed.), Univ. of Maryland, College Park, 1971. pp. 547-583. MR 0287723 (44:4926)
  • [SF] G. Strang and G. Fix, A Fourier analysis of the finite element variational melhod, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), C.I.M.E., 1973, pp. 793-840.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A25, 65N30

Retrieve articles in all journals with MSC: 41A25, 65N30

Additional Information

Keywords: Controlled approximation, approximation order, multivariate, box splines, finite element analysis, Fourier series
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society