Accelerating convergence of trigonometric approximations

Authors:
William B. Jones and G. Hardy

Journal:
Math. Comp. **24** (1970), 547-560

MSC:
Primary 65.20; Secondary 42.00

DOI:
https://doi.org/10.1090/S0025-5718-1970-0277086-X

MathSciNet review:
0277086

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Lanczos has recently developed a method for accelerating the convergence of trigonometric approximations for smooth, nonperiodic functions by modifying their boundary behavior. The method is reformulated here in terms of interpolation theory and is shown to be related to the theory of Lidstone interpolation. Extensions given include a new type of modifying function and the establishment of criteria for the convergence of associated interpolation series. Applications are given for the error function and its derivative.

**[1]**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****[2]**Ralph P. Boas Jr. and R. Creighton Buck,*Polynomial expansions of analytic functions*, Second printing, corrected. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Bd. 19, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin, 1964. MR**0162914****[3]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[4]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[5]**Dunham Jackson,*The theory of approximation*, American Mathematical Society Colloquium Publications, vol. 11, American Mathematical Society, Providence, RI, 1994. Reprint of the 1930 original. MR**1451140****[6]**William B. Jones & G. Hardy,*Numerical Prediction of Ionospheric Characteristics*, ESSA Technical Report ERL 76-ITS 66, U.S. Government Printing Office, Washington, D.C., 1968.**[7]**William B. Jones & F. Stewart,*Numerical Mapping of Ionospheric Plasma Frequency*, paper presented at URSI Spring Meeting in Washington, D.C., 1969.**[8]**C. Lanczos,*Evaluation of noisy data*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**1**(1964), 76–85. MR**0179909****[9]**Cornelius Lanczos,*Discourse on Fourier series*, Hafner Publishing Co., New York, 1966. MR**0199629****[10]**G. J. Lidstone, "Notes on the extension of Aitken's theorem (for polynomial interpolation) to the Everett types,"*Proc. Edinburgh Math. Soc.*(2), v. 2, 1929, pp. 16-19.**[11]**J. M. Whittaker, "On Lidstone series and two-point expansions of analytic functions,"*Proc. London Math. Soc.*, v. 36, 1934, pp. 451-469.**[12]**D. V. Widder,*Completely convex functions and Lidstone series*, Trans. Amer. Math. Soc.**51**(1942), 387–398. MR**0006356**, https://doi.org/10.1090/S0002-9947-1942-0006356-4

Retrieve articles in *Mathematics of Computation*
with MSC:
65.20,
42.00

Retrieve articles in all journals with MSC: 65.20, 42.00

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0277086-X

Keywords:
Trigonometric approximation,
interpolation series,
accelerating convergence,
Fourier series,
Lidstone interpolation

Article copyright:
© Copyright 1970
American Mathematical Society