Computational techniques based on the Lanczos representation
J. N. Lyness
Math. Comp. 28 (1974), 81-123
Primary 65D15; Secondary 42A08
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Abstract: In his book Discourse on Fourier Series, Lanczos deals in some detail with representations of of the type where is a polynomial of degree and has the property that its full range Fourier coefficients converge at the rate .
In Part I, some properties of and of the series are described. These properties are used here to provide criteria for the convergence or divergence of the Euler-Maclaurin series, in the case when is an analytic function. The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned.
In Part II, the Lanczos representation is employed to derive an approximate representation for an analytic function on the interval [0, 1] is derived. This has the form and requires for its determination the values of the derivatives and the regularly spaced function values . It involves replacing by a discrete Fourier expansion based on trapezoidal rule approximations to its Fourier coefficients.
This representation is a powerful one. The drawback is that it requires derivatives. Most of Part II is devoted to the effect of using only approximate derivatives. It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable. In a computational context the only penalty for using inaccurate derivatives is that a larger value of m may--or may not--be required to attain a specific accuracy.
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- C. Lanczos, Discourse on Fourier Series, Hafner, New York, 1966. MR 33 #7772. MR 0199629 (33:7772)
- 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.
- 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)
- P. C. Chakravarti, Integrals and Sums, Univ. of London, The Athlone Press, 1970.
- W. M. Gentleman & G. Sande, Fast Fourier Transforms--For Fun and Profit, Proc. AF1PS 1966 FJCC, v. 29, Spartan Books, New York, pp. 563-578.
- J. N. Lyness & G. Sande, "ENTCAF and ENTCRE: Evaluation of normalised Taylor coefficients of an analytic function, algorithm 413," Comm. ACM, v. 14, 1971, pp. 669-675.
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