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.

In this paper we introduce a set of orthonormal functions, , where ϕn[r] is composed of a sine function and a sigmoidal transformation γr of order r>0. Based on the proposed functions ϕn[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 ℝ and then, by reconstructing the existing sigmoidal cosine functions ψn[r] and the presented functions ϕn[r], we develop two kinds of 2-periodic series expansions on ℝ. Superiority of the presented sigmoidal-type series in approximating a function by the partial sum is demonstrated by numerical examples.