A method for the approximation of functions defined by formal series expansions in orthogonal polynomials

Author:
Jonas T. Holdeman

Journal:
Math. Comp. **23** (1969), 275-287

MSC:
Primary 41.30; Secondary 42.00

DOI:
https://doi.org/10.1090/S0025-5718-1969-0251412-1

MathSciNet review:
0251412

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Abstract: An algorithm is described for numerically evaluating functions defined by formal (and possibly divergent) series as well as convergent series of orthogonal functions which are, apart from a factor, orthogonal polynomials. When the orthogonal functions are polynomials, the approximations are rational functions. The algorithm is similar in some respects to the method of Padé approximants. A rational approximation involving Tchebychev polynomials due to H. Maehley and described by E. Kogbetliantz [1] is a special case of the algorithm.

**[1]**E. G. Kogbetliantz, ``Generation of elementary functions,'' in*Mathematical Methods for Digital Computers*, Wiley, New York, 1960. MR**22**#8681. MR**0117907 (22:8681)****[2]**A. Erdélyi, et al.,*Higher Transcendental Functions*, Vol. II, McGraw-Hill, New York, 1953. MR**15**, 419.**[3]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR**1**, 14.**[4]**J. Holdeman, ``Legendre polynomial expansions of hypergeometric functions with applications.'' (To appear.) MR**0254279 (40:7488)****[5]**G. A. Baker, Jr., J. L. Gammel & J. G. Wills, ``An investigation of the applicability of the Padé approximant method,''*J. Math. Anal. Appl.*, v. 2, 1961, pp. 405-418. MR**23**#B3125. MR**0130093 (23:B3125)****[6]**G. A. Baker, Jr., ``The theory and application of the Padé approximant method,'' in*Advances in Theoretical Physics*, Vol. I, Academic Press, New York, 1965, pp. 1-58. MR**32**#5253. MR**0187807 (32:5253)**

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DOI:
https://doi.org/10.1090/S0025-5718-1969-0251412-1

Article copyright:
© Copyright 1969
American Mathematical Society