Rational Trigonometric Approximations Using Fourier Series Partial Sums
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING HAMPTON VA
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A class of approximations SN,M to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied.. Each approximation SN,M is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of SN,M agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these Fourier-Pad4 approximations converge point-wise to fxfx-l2 more rapidly in some cases by a factor of 1k2M than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented. Fourier series, Rational approximations, Gibbs phenomena.
- Numerical Mathematics