Resumen de Sigmoidal cosine series on the interval
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.
