International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 1, Pages 33-42
doi:10.1155/IJMMS.2005.33
Corrected Fourier series and its application to function approximation
1Key Laboratory of Marine Science and Numerical Modeling, The First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China
2Department of Oceanography, University of Hawaii at Manoa, 1000 Pope Road, Honolulu 96822, HI, USA
Received 30 November 2003; Revised 9 June 2004
Copyright © 2005 Qing-Hua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Any quasismooth function f(x)
in a finite interval [0,x0], which has only a finite number of finite discontinuities and has
only a finite number of extremes, can be approximated by a
uniformly convergent Fourier series and a correction function. The
correction function consists of algebraic polynomials and
Heaviside step functions and is required by the aperiodicity at
the endpoints (i.e., f(0)≠f(x0)) and the finite discontinuities in between.
The uniformly convergent Fourier series
and the correction function are collectively referred to as the
corrected Fourier series. We prove that in order for the
mth derivative of the Fourier series to be uniformly
convergent, the order of the polynomial need not exceed (m+1). In other words, including the no-more-than-(m+1) polynomial has
eliminated the Gibbs phenomenon of the Fourier series until its
mth derivative. The corrected Fourier series is then applied to
function approximation; the procedures to determine the
coefficients of the corrected Fourier series are illustrated in detail using examples.