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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
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.
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Abramowitz and Irene
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P. Boas Jr. and R.
Creighton Buck, Polynomial expansions of analytic functions,
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0162914 (29 #218)
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Eugene
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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.
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Lanczos, Evaluation of noisy data, J. Soc. Indust. Appl. Math.
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(33 #7772)
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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.
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J. M. Whittaker, "On Lidstone series and two-point expansions of analytic functions," Proc. London Math. Soc., v. 36, 1934, pp. 451-469.
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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/S0002-9947-1942-0006356-4
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- 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)
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- E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 34 #924. MR 0201039 (34:924)
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- 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 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., v. 1, 1964, pp. 76-85. MR 31 #4146. MR 0179909 (31:4146)
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- 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. 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., v. 51, 1942, pp. 387-398. MR 3, 293. MR 0006356 (3:293b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025-5718-1970-0277086-X
PII:
S 0025-5718(1970)0277086-X
Keywords:
Trigonometric approximation,
interpolation series,
accelerating convergence,
Fourier series,
Lidstone interpolation
Article copyright:
© Copyright 1970 American Mathematical Society
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