Coefficients in series expansions for certain classes of functions

Authors:
P. D. Tuan and David Elliott

Journal:
Math. Comp. **26** (1972), 213-232

MSC:
Primary 42A52; Secondary 65D99

DOI:
https://doi.org/10.1090/S0025-5718-1972-0301440-2

MathSciNet review:
0301440

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of evaluating or estimating the coefficients for the expansion of a function in a series of classical orthogonal polynomials is examined. By restricting the functions under consideration to the classes of integral transforms and inverse integral transforms, the coefficients may be expressed in alternative forms which often are more amenable to analysis.

- G. Sansone,
*Orthogonal functions*, Pure and Applied Mathematics, Vol. IX, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1959. Revised English ed; Translated from the Italian by A. H. Diamond; with a foreword by E. Hille. MR**0103368**
J. Wimp, “Polynomial approximations to integral transforms,” - Y. L. Luke and J. Wimp,
*Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray*, Math. Comp.**17**(1963), 395–404. MR**157014**, DOI https://doi.org/10.1090/S0025-5718-1963-0157014-4 - David Elliott,
*The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function*, Math. Comp.**18**(1964), 274–284. MR**166903**, DOI https://doi.org/10.1090/S0025-5718-1964-0166903-7 - David Elliott and George Szekeres,
*Some estimates of the coefficients in the Chebyshev series expansion of a function*, Math. Comp.**19**(1965), 25–32. MR**172447**, DOI https://doi.org/10.1090/S0025-5718-1965-0172447-X - G. F. Miller,
*On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation*, SIAM J. Numer. Anal.**3**(1966), 390–409. MR**203312**, DOI https://doi.org/10.1137/0703034 - Jet Wimp,
*The asymptotic representation of a class of $G$-functions for large parameter*, Math. Comp.**21**(1967), 639–646. MR**223617**, DOI https://doi.org/10.1090/S0025-5718-1967-0223617-5
Y. L. Luke, - L. J. Slater,
*Confluent hypergeometric functions*, Cambridge University Press, New York, 1960. MR**0107026** - D. S. Jones,
*Generalised functions*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0217534** - Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642** - Wilbur R. LePage,
*Complex variables and the Laplace transform for engineers*, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1961. MR**0117514**
A. Erdélyi et al., - David Vernon Widder,
*The Laplace Transform*, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR**0005923** - N. G. de Bruijn,
*Asymptotic methods in analysis*, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1958. MR**0099564**

*Math. Comp.*, v. 15, 1961, pp. 174-178. C. W. Clenshaw,

*Chebyshev Series for Mathematical Functions*, National Physical Lab. Math. Tables, vol. 5, Department of Scientific and Industrial Research, HMSO, London, 1962. MR

**26**#362.

*The Special Functions and their Approximations*. Vol. 1, Academic Press, New York and London, 1969. MR

**39**#3039. A. Erdélyi et al.,

*Tables of Integral Transforms*. Vol. 1, McGraw-Hill, New York, 1954. MR

**15**, 868. A. Erdéyi et al.,

*Tables of Integral Transforms*. Vol. 2, McGraw-Hill, New York, 1954. MR

**16**, 468. E. C. Titchmarsh,

*Introduction to the Theory of Fourier Integrals*, Clarendon Press, Oxford, 1937. P. D. Tuan,

*On the Estimation of Fourier Coefficients*, Ph.D. Thesis, University of Tasmania, Tasmania, 1969.

*Higher Transcendental Functions*. Vol. 2, McGraw-Hill, New York, 1953. MR

**15**, 419. A. Erdélyi et al.,

*Higher Transcendental Functions*. Vol. 1, McGraw-Hill, New York, 1953. MR

**15**, 419. E. C. Titchmarsh,

*The Theory of Functions*, Clarendon Press, Oxford, 1932. S. Saks,

*Theory of the Integral*, Monografie Mat., vol. 7, PWN, Warsaw, 1937.

Retrieve articles in *Mathematics of Computation*
with MSC:
42A52,
65D99

Retrieve articles in all journals with MSC: 42A52, 65D99

Additional Information

Keywords:
Coefficients in series expansions of functions,
Jacobi polynomials,
Fourier transform,
generalised function,
Laplace transform,
Stieltjes transform

Article copyright:
© Copyright 1972
American Mathematical Society