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

Full-text PDF

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 (Unpublished.)**[2]**H. C. Thacher, Jr.,*Inductive Interpolation Algorithm*, 1963. (Unpublished.)**[3]**A. C. R. Newbery,*Interpolation by algebraic and trigonometric polynomials*, Math.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.**20**(1966), 597–599. MR**0203905**, https://doi.org/10.1090/S0025-5718-1966-0203905-8**[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]**Martin Davis,*Computability and unsolvability*, McGraw-Hill Series in Information Processing and Computers, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958. MR**0124208****[6]**J. F. Steffensen,*Interpolation*, Chelsea Publishing Co., New York, N. Y., 1950. 2d ed. MR**0036799****[7]**W. M. Kincaid,*Note on the error in interpolation of a function of two independent variables*, Ann. Math. Statistics**19**(1948), 85–88. MR**0024231****[8]**Arthur Sard,*Remainders: functions of several variables*, Acta Math.**84**(1951), 319–346. MR**0040354**, https://doi.org/10.1007/BF02414860**[9]**Henry C. Thacher Jr.,*Derivation of interpolation formulas in several independent variables.*, Ann. New York Acad. Sci.**86**(1960), 758–775 (1960). MR**0116456**

Retrieve articles in *Mathematics of Computation*
with MSC:
65D05,
41A63

Retrieve articles in all journals with MSC: 65D05, 41A63

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