Accelerating convergence of trigonometric approximations
Authors:
William B. Jones and G. Hardy
Journal:
Math. Comp. 24 (1970), 547560
MSC:
Primary 65.20; Secondary 42.00
MathSciNet review:
0277086
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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
(29 #4914)
 [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;
SpringerVerlag, Berlin, 1964. MR 0162914
(29 #218)
 [3]
Philip
J. Davis, Interpolation and approximation, Blaisdell
Publishing Co. Ginn and Co. New YorkTorontoLondon, 1963. MR 0157156
(28 #393)
 [4]
Eugene
Isaacson and Herbert
Bishop Keller, Analysis of numerical methods, John Wiley &
Sons, Inc., New YorkLondonSydney, 1966. MR 0201039
(34 #924)
 [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
(98a:01022)
 [6]
William B. Jones & G. Hardy, Numerical Prediction of Ionospheric Characteristics, ESSA Technical Report ERL 76ITS 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
(31 #4146)
 [9]
Cornelius
Lanczos, Discourse on Fourier series, Hafner Publishing Co.,
New York, 1966. MR 0199629
(33 #7772)
 [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. 1619.
 [11]
J. M. Whittaker, "On Lidstone series and twopoint expansions of analytic functions," Proc. London Math. Soc., v. 36, 1934, pp. 451469.
 [12]
D.
V. Widder, Completely convex functions and
Lidstone series, Trans. Amer. Math. Soc. 51 (1942), 387–398.
MR
0006356 (3,293b), http://dx.doi.org/10.1090/S00029947194200063564
 [1]
 M. Abramowitz & I. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series, 55, Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964, chap. 7. MR 29 #4914. MR 0167642 (29:4914)
 [2]
 R. P. Boas, Jr. & R. C. Buck, Polynomial Expansions of Analytic Functions, 2nd rev. ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 19, Academic Press, New York and SpringerVerlag, Berlin and New York, 1964. MR 29 #218. MR 0162914 (29:218)
 [3]
 Philip J. Davis, Interpolation and Approximation, Blaisdell, Waltham, Mass., 1963, pp. 2830. MR 28 #393. MR 0157156 (28:393)
 [4]
 E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 #924. MR 0201039 (34:924)
 [5]
 D. Jackson, The Theory of Approximation, Amer. Math. Soc. Colloq. Publ., vol. XI, Amer. Math. Soc., Providence, R.I., 1930. MR 1451140 (98a:01022)
 [6]
 William B. Jones & G. Hardy, Numerical Prediction of Ionospheric Characteristics, ESSA Technical Report ERL 76ITS 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., v. 1, 1964, pp. 7685. MR 31 #4146. MR 0179909 (31:4146)
 [9]
 C. Lanczos, Discourse on Fourier Series, Hafner, New York, 1966. MR 33 #7772. MR 0199629 (33:7772)
 [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. 1619.
 [11]
 J. M. Whittaker, "On Lidstone series and twopoint expansions of analytic functions," Proc. London Math. Soc., v. 36, 1934, pp. 451469.
 [12]
 D. V. Widder, "Completely convex functions and Lidstone series," Trans. Amer. Math. Soc., v. 51, 1942, pp. 387398. MR 3, 293. MR 0006356 (3:293b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819700277086X
PII:
S 00255718(1970)0277086X
Keywords:
Trigonometric approximation,
interpolation series,
accelerating convergence,
Fourier series,
Lidstone interpolation
Article copyright:
© Copyright 1970
American Mathematical Society
