Sigmoidal type series expansion

Abstract

In this paper we introduce a set of orthonormal functions, $\{\phi_n^{[r]}\}_{n=1}^{\infty}$, where $\phi_n^{[r]}$ is composed of a sine function and a sigmoidal transformation $\gm_r$ of order $r>0$. Based on the present functions $\phi_n^{[r]}$ named by sigmoidal sine functions, we consider a series expansion of a function on the interval $[-1,1]$ and the related convergence analysis. Furthermore, we extend the sigmoidal transformation to the whole real line $\mathbf{R}$ and then by reconstructing the existing sigmoidal cosine functions $\psi_n^{[r]}$ and the present functions $\phi_n^{[r]}$, we develop two kinds of 2-periodic series expansion on $\mathbf{R}$. Superiority of the presented sigmoidal type series in approximating a function by the partial sum is demonstrated by numerical examples. doi:10.1017/S1446181108000060