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Sigmoidal cosine series on the interval

Published online by Cambridge University Press:  17 February 2009

Beong In Yun
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.

Keywords

Fourier seriessigmoidal transformationsigmoidal cosine series.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 4 , April 2006, pp. 451 - 475
Copyright
Copyright © Australian Mathematical Society 2006

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