Rate of convergence of Shepard's global interpolation formula

Author:
Reinhard Farwig

Journal:
Math. Comp. **46** (1986), 577-590

MSC:
Primary 65D05; Secondary 41A05, 41A25

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829627-0

MathSciNet review:
829627

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given any data points in and values of a function *f*, Shepard's global interpolation formula reads as follows:

*f*of degree . In this paper, we investigate the approximating power of for all values of

*p*,

*q*and

*s*.

**[1]**A. B. Bash-Ayan,*Algorithms for the Interpolation of Scattered Data on the Plane*, Thesis, Brighton Polytechnic, 1983.**[2]**Robert E. Barnhill,*Representation and approximation of surfaces*, Mathematical software, III (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1977) Academic Press, New York, 1977, pp. 69–120. Publ. Math. Res. Center Univ. Wisconsin, No. 39. MR**0489081****[3]**R. E. Barnhill, R. P. Dube, and F. F. Little,*Properties of Shepard’s surfaces*, Rocky Mountain J. Math.**13**(1983), no. 2, 365–382. MR**702831**, https://doi.org/10.1216/RMJ-1983-13-2-365**[4]**Richard Franke,*Scattered data interpolation: tests of some methods*, Math. Comp.**38**(1982), no. 157, 181–200. MR**637296**, https://doi.org/10.1090/S0025-5718-1982-0637296-4**[5]**William J. Gordon and James A. Wixom,*Shepard’s method of “metric interpolation” to bivariate and multivariate interpolation*, Math. Comp.**32**(1978), no. 141, 253–264. MR**0458027**, https://doi.org/10.1090/S0025-5718-1978-0458027-6**[6]**D. H. McLain, "Drawing contours from arbitrary data points,"*Comput. J.*, v. 17, 1974, pp. 318-324.**[7]**P. Lancaster and K. Salkauskas,*Surfaces generated by moving least squares methods*, Math. Comp.**37**(1981), no. 155, 141–158. MR**616367**, https://doi.org/10.1090/S0025-5718-1981-0616367-1**[8]**Robert E. Barnhill and Wolfgang Boehm (eds.),*Surfaces in computer aided geometric design*, North-Holland Publishing Co., Amsterdam-New York, 1983. MR**709285****[9]**D. J. Newman and T. J. Rivlin,*Optimal universally stable interpolation*, Analysis**3**(1983), no. 1-4, 355–367. MR**756124****[10]**C. C. Poeppelmeier,*A Boolean Sum Interpolation Scheme to Random Data for Computer Aided Geometric Design*, Thesis, University of Utah, 1975.**[11]**Larry L. Schumaker,*Fitting surfaces to scattered data*, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 203–268. MR**0426369****[12]**D. Shepard,*A Two-Dimensional Interpolation Function for Irregularly Spaced Data*, Proc. 23rd Nat. Conf. ACM, 1968, pp. 517-524.**[13]**Hassler Whitney,*Functions differentiable on the boundaries of regions*, Ann. of Math. (2)**35**(1934), no. 3, 482–485. MR**1503174**, https://doi.org/10.2307/1968745

Retrieve articles in *Mathematics of Computation*
with MSC:
65D05,
41A05,
41A25

Retrieve articles in all journals with MSC: 65D05, 41A05, 41A25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0829627-0

Keywords:
Multivariate interpolation,
Shepard's formula,
rate of convergence

Article copyright:
© Copyright 1986
American Mathematical Society