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Sigmoidal cosine series on the interval

Published online by Cambridge University Press:  17 February 2009

Beong In Yun
Affiliation:
Faculty of Mathematics, Informatics and Statistics, Kunsan National University, 573–701, Korea; e-mail: biyun@kunsan.ac.kr or paulll@maths.uq.edu.au.
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Abstract

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We construct a set of functions, say, composed of a cosine function and a sigmoidal transformation γr of order r > 0. The present functions are orthonormal with respect to a proper weight function on the interval [−1, 1]. It is proven that if a function f is continuous and piecewise smooth on [−1, 1] then its series expansion based on converges uniformly to f so long as the order of the sigmoidal transformation employed is 0 < r ≤ 1. Owing to the variational feature of according to the value of r, one can expect improvement of the traditional Fourier series approximation for a function on a finite interval. Several numerical examples show the efficiency of the present series expansion in comparison with the Fourier series expansion.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2006

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