Converting interpolation series into Chebyshev series by recurrence formulas
Author: Herbert E. Salzer
Journal: Math. Comp. 30 (1976), 295-302
MSC: Primary 65D05; Secondary 65Q05
MathSciNet review: 0395159
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Abstract: Interpolation series (divided difference, Gregory-Newton, Gauss, Stirling, Bessel) are converted into Chebyshev (or Jacobi) series by applying a previously derived general five-term recurrence formula . It employs the coefficients in three-term linear recurrence formulas (same kind as for orthogonal polynomials) which have been found for the mth degree nonorthogonal polynomial coefficients of the differences used in the interpolation series. In the Gauss, Stirling and Bessel series, the coefficients in the recurrence formulas vary with the parity of m. The basic five-term recurrence formula is applicable also to: (1) inter- and intraconversion of power series in $ax + b$, divided difference and equal-interval interpolation series (including subtabulation), and Chebyshev series, (2) obtaining Chebyshev series for solutions of difference equations, (3) the derivation of formulas for Chebyshev coefficients in terms of differences, and (4) the conversion of interpolation series into Chebyshev series, for more than one variable.
- J. C. P. Miller, Two numerical applications of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh Sect. A 62 (1946), 204–210. MR 17583
- L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan and Co., Ltd., London, 1951. MR 0043339
- Herbert E. Salzer, A recurrence scheme for converting from one orthogonal expansion into another, Comm. ACM 16 (1973), no. 11, 705–707. MR 0395158, DOI https://doi.org/10.1145/355611.362548
J. C. P. MILLER, "Two numerical applications of Chebyshev polynomials," Proc. Roy. Soc. Edinburgh Sect. A, v. 62, 1946, pp. 204-210. MR 8, 172.
L. M. MILNE-THOMSON, Calculus of Finite Differences, Macmillan, London, 1933, pp. 1-7, 55-70.
H. E. SALZER, "A recurrence scheme for converting from one orthogonal expansion into another," Comm. ACM, v. 16, 1973, pp. 705-707.
Keywords: Series interconversion, interpolation series, Chebyshev series, recurrence formulas, three-term recurrence, five-term recurrence, difference equations
Article copyright: © Copyright 1976 American Mathematical Society