## Cardinal interpolation by multivariate splines

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- by C. K. Chui, K. Jetter and J. D. Ward PDF
- Math. Comp.
**48**(1987), 711-724 Request permission

## Abstract:

The purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials. In particular, the notion of a commutator is introduced and its connection with the Marsden identity is observed. The order of a commutator is shown to be equivalent to the Strang and Fix conditions that arise in the study of the local approximation orders using quasi-interpolants. We also prove that scaled cardinal interpolants give these local approximation orders.## References

- Shmuel Agmon,
*Lectures on elliptic boundary value problems*, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR**0178246** - C. de Boor and K. Höllig,
*$B$-splines from parallelepipeds*, J. Analyse Math.**42**(1982/83), 99–115. MR**729403**, DOI 10.1007/BF02786872 - Carl de Boor, Klaus Höllig, and Sherman Riemenschneider,
*Bivariate cardinal interpolation by splines on a three-direction mesh*, Illinois J. Math.**29**(1985), no. 4, 533–566. MR**806466** - 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**, DOI 10.1090/S0002-9939-1985-0810161-X - J. H. Bramble and S. R. Hilbert,
*Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation*, SIAM J. Numer. Anal.**7**(1970), 112–124. MR**263214**, DOI 10.1137/0707006 - Wolfgang Dahmen and Charles A. Micchelli,
*Translates of multivariate splines*, Linear Algebra Appl.**52/53**(1983), 217–234. MR**709352**, DOI 10.1016/0024-3795(83)80015-9 - Wolfgang Dahmen and Charles A. Micchelli,
*Recent progress in multivariate splines*, Approximation theory, IV (College Station, Tex., 1983) Academic Press, New York, 1983, pp. 27–121. MR**754343** - 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**, DOI 10.1017/S033427000000446X
P. O. Frederickson, - Walter Rudin,
*Functional analysis*, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR**0365062** - I. J. Schoenberg,
*Cardinal spline interpolation*, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR**0420078** - I. J. Schoenberg,
*Cardinal interpolation and spline functions*, J. Approximation Theory**2**(1969), 167–206. MR**257616**, DOI 10.1016/0021-9045(69)90040-9 - I. J. Schoenberg,
*Cardinal interpolation and spline functions. II. Interpolation of data of power growth*, J. Approximation Theory**6**(1972), 404–420. MR**340899**, DOI 10.1016/0021-9045(72)90048-2 - Larry L. Schumaker,
*Spline functions: basic theory*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR**606200**
G. Strang & G. Fix, "A Fourier analysis of the finite element variational method," C.I.M.E. II Cilo 1971,

*Generalized Triangular Splines*, Mathematics Report 7-71, Lakehead Univ., 1971.

*Constructive Aspects of Functional Analysis*(G. Geymonat, ed.), 1973, pp. 793-840.

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp.
**48**(1987), 711-724 - MSC: Primary 41A05; Secondary 41A15, 41A63
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878701-2
- MathSciNet review: 878701