ANZIAM  J.  47 (2006), 451-475
Sigmoidal cosine series on the interval

 Beong In Yun   Faculty of Mathematics   Informatics and Statistics   Kunsan National University   573--701   Korea  biyun@kunsan.ac.kr or paulll@maths.uq.edu.au

Abstract
We construct a set of functions, say, composed of a cosine function and a sigmoidal transformation of order . The present functions are orthonormal with respect to a proper weight function on the interval . It is proven that if a function is continuous and piecewise smooth on then its series expansion based on converges uniformly to so long as the order of the sigmoidal transformation employed is . 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.