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

Beong In Yun
  Faculty of Mathematics
  Informatics and Statistics
  Kunsan National University

We construct a set of functions, say, $\psi_n^{[r]}$ composed of a cosine function and a sigmoidal transformation $\gamma_r$ 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 $\psi_n^{[r]}$ converges uniformly to $f$ so long as the order of the sigmoidal transformation employed is $0<r\le1$. Owing to the variational feature of $\psi_n^{[r]}$ 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.
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