Generalized recursive multivariate interpolation

Author:
Earl H. McKinney

Journal:
Math. Comp. **26** (1972), 723-735

MSC:
Primary 65D05; Secondary 41A63

DOI:
https://doi.org/10.1090/S0025-5718-1972-0341797-X

MathSciNet review:
0341797

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Abstract | References | Similar Articles | Additional Information

Abstract: A generalized recursive interpolation technique for a set of linear functionals over a set of general univariate basis functions has been previously developed. This paper extends these results to restricted multivariate interpolation over a set of general multivariate basis functions. When the data array is a suitable configuration (e.g., an -dimensional simplex), minimal degree multivariate interpolating polynomials are produced by this recursive interpolation scheme. By using product rules, recursive univariate interpolation applied to each variable singly produces multivariate interpolating polynomials (not of minimal degree) when the data are arranged in a hyper-rectangular array. By proper ordering of points in a data array, multivariate polynomial interpolation is accomplished over other arrays such as diamonds and truncated diamonds in two dimensions and their counterparts in dimensions.

**[1]**P. L. Walker,*Generalized Recursive Interpolation*, Thesis, University of Kansas, Lawrence, Kan., 1964. (Unpublished.)**[2]**H. C. Thacher, Jr.,*Inductive Interpolation Algorithm*, 1963. (Unpublished.)**[3]**A. C. R. Newbery, ``Interpolation by algebraic and trigonometric polynomials,''*Math. Comp.*, v. 20, 1966, pp. 597-599. MR**34**#3752. MR**0203905 (34:3752)****[4]**W. E. Milne, W. Arntzen, N. Reynolds & J. Wheelock,*Mathematics for Digital Computers*. Vol. I.*Multivariate Interpolation*, WADC Technical Report 57-556, 1958.**[5]**M. Davis,*Computability and Unsolvability*, McGraw-Hill Series in Information Processing and Computers, McGraw-Hill, New York, 1958, pp. 43-46. MR**23**#A1525. MR**0124208 (23:A1525)****[6]**J. F. Steffensen,*Interpolation*, 2nd ed., Chelsea, New York, 1950. MR**12**, 164. MR**0036799 (12:164d)****[7]**W. M. Kincaid, ``Note on the error in interpolation of a function of two independent variables,''*Ann. Math. Statist.*, v. 19, 1948, pp. 85-88. MR**9**, 470; MR**10**, 55. MR**0024231 (9:470i)****[8]**A. Sard, ``Remainders: Functions of several variables,''*Acta Math.*, v. 84, 1951, pp. 319-346. MR**12**, 680. MR**0040354 (12:680c)****[9]**H. C. Thacher, Jr., ``Derivation of interpolation formulas in several independent variables,''*Ann. New York Acad. Sci.*, v. 86, 1960, pp. 758-775. MR**22**#7243. MR**0116456 (22:7243)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0341797-X

Keywords:
Recursive multivariate interpolation,
repeated recursive univariate interpolation,
hyper-rectangular array,
basis functions

Article copyright:
© Copyright 1972
American Mathematical Society