   ISSN 1088-6842(online) ISSN 0025-5718(print)

A general method of approximation. I

Authors: Staffan Wrigge and Arne Fransén
Journal: Math. Comp. 38 (1982), 567-588
MSC: Primary 41A50; Secondary 15A57, 65D15
DOI: https://doi.org/10.1090/S0025-5718-1982-0645672-9
MathSciNet review: 645672
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Abstract: In this paper we study two families of functions, viz. F and H, and show how to approximate the functions considered in the interval [0,1 ]. The functions are assumed to be real when the argument is real. We define $F = \{ f;({\text {i}}) f\left ( {\frac {1}{2} + x} \right ) = f\left ( {\frac {1}{2} - x} \right ),({\text {ii}}) f(0) = f(1) = 0,({\text {iii}})\;f(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\},$ $H = \{ h;({\text {j}})\;h\left ( {\frac {1}{2} + x} \right ) = - h\left ( {\frac {1}{2} - x} \right ),({\text {jj}})\;h(0) = h(1) = 0,({\text {jjj}})\;h(x)\;{\text {is analytic in a sufficiently large neighborhood of}}\;x = 0\}.$ The approximations are defined in the interval [0,1 ] by $\min \int _0^1 {{{\left ( {f(x) - \sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx}$ and $\min \int _0^1 {{{\left ( {h(x) - (1 - 2x)\sum \limits _{n = 1}^k {{c_{n,k}}{{[x(1 - x)]}^n}} } \right )}^2}{x^q}{{(1 - x)}^q}\;dx} ,$ where $q \in \{ 0,1,2, \ldots \}$. The associated matrices are analyzed using the theory of orthogonal polynomials, especially the Jacobi polynomials ${G_n}(p,q,x)$. We apply the general theory to the basic trigonometric functions $\sin (x)$ and $\cos (x)$.

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• 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
• F. Ayres, Jr., Theory and Problems of Matrices, Schaum Publishing Company, 1962. H. Cramér, Mathematical Methods of Statistics, 10th ed., Princeton Univ. Press, Princeton, N. J., 1963.
• L. A. Lyusternik, O. A. Chervonenkis, and A. R. Yanpol′skii, Handbook for computing elementary functions, Pergamon Press, Oxford-Edinburg-New York, 1965. Translated from the Russian by G. J. Tee; Translation edited by K. L. Stewart. MR 0183102
• Richard Savage and Eugene Lukacs, Tables of inverses of finite segments of the Hilbert matrix, Contributions to the solution of systems of linear equations and the determination of eigenvalues, National Bureau of Standards Applied Mathematics Series No. 39, U. S. Government Printing Office, Washington, D. C., 1954, pp. 105–108. MR 0068303
• S. Wrigge, A. Fransén & G. Borenius, Rapid Calculation of $\sin (x)$, FOA Rapport, C 10150-M8, National Defence Research Institute, S 104 50 Stockholm 80, Sweden, 1980. S. Wrigge, A. Fransén & G. Borenius, A General Method of Approximation, Particularly in ${L_2}$, FOA Rapport C 10158-M8, National Defence Research Institute, S 104 50 Stockholm 80, Sweden, 1980. S. Wrigge, A General Method of Approximation Associated with Bernstein Polynomials, FOA Rapport, C 10170-M8, National Defence Research Institute, S 104 50 Stockholm 80, Sweden, 1980.

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