A natural formulation of quasi-interpolation by multivariate splines

Authors:
Charles K. Chui and Harvey Diamond

Journal:
Proc. Amer. Math. Soc. **99** (1987), 643-646

MSC:
Primary 41A15; Secondary 41A65

MathSciNet review:
877032

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Abstract: A quasi-interpolation formula based on discrete function values is given in the form of a multivariate spline series that yields the local approximation order characterized by the Fix-Strang conditions. This formula can be considered as a partial sum of the formal Neumann series expansion of the formal interpolation operator, and hence, justifies that "quasi-interpolation" is indeed an appropriate terminology for such an approximation formula.

**[1]**C. de Boor and G. J. Fix,*Spline approximation by quasiinterpolants*, J. Approximation Theory**8**(1973), 19–45. Collection of articles dedicated to Isaac Jacob Schoenberg on his 70th birthday, I. MR**0340893****[2]**C. de Boor and R.-Q. Jia,*Controlled approximation and a characterization of the local approximation order*, Proc. Amer. Math. Soc.**95**(1985), no. 4, 547–553. MR**810161**, 10.1090/S0002-9939-1985-0810161-X**[3]**C. K. Chui, K. Jetter, and J. D. Ward,*Cardinal interpolation by multivariate splines*, Math. Comp.**48**(1987), no. 178, 711-724. MR**878701**, 10.1090/S0025-5718-1987-0878701-2**[4]**C. K. Chui and M. J. Lai,*A multivariate analog of Marsden's identity and a quasi-interpolation scheme*, J. Constructive Approx. (to appear).**[5]**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**[6]**G. Strang and G. Fix,*Fourier analysis of the finite element variational method*, C.I.M.E. II Cilo 1971, Constructive Aspects of Functionl Analysis (G. Geymonat, Ed.), 1973, pp. 793-840.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1987-0877032-6

Keywords:
Multivariate splines,
local approximation order,
quasi-interpolation,
Neumann series

Article copyright:
© Copyright 1987
American Mathematical Society