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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The local approximation order from a scale of approximating functions on is characterized in terms of the linear span (and its Fourier transform) of the finitely many compactly supported functions whose integer translates , span the space from which the scale is derived. This provides a correction of similar results stated and proved, in part, by Strang and Fix.

**[BH]**C. de Boor and K. Höllig,*𝐵-splines from parallelepipeds*, J. Analyse Math.**42**(1982/83), 99–115. MR**729403**, 10.1007/BF02786872**[DM]**Wolfgang Dahmen and Charles A. Micchelli,*On the approximation order from certain multivariate spline spaces*, J. Austral. Math. Soc. Ser. B**26**(1984), no. 2, 233–246. MR**765640**, 10.1017/S033427000000446X**[FS]**George Fix and Gilbert Strang,*Fourier analysis of the finite element method in Ritz-Galerkin theory.*, Studies in Appl. Math.**48**(1969), 265–273. MR**0258297****[J]**R.-q. Jia,*A counterexample to a result of Strang and Fix concerning controlled approximation*, MRC TSR# 2743, 1984.**[R]**Walter Rudin,*Function theory in the unit ball of 𝐶ⁿ*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594****[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]**Gilbert Strang,*The finite element method and approximation theory*, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 547–583. MR**0287723****[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.

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0810161-X

Keywords:
Controlled approximation,
approximation order,
multivariate,
box splines,
finite element analysis,
Fourier series

Article copyright:
© Copyright 1985
American Mathematical Society