Article contents
Sigmoidal cosine series on the interval
Published online by Cambridge University Press: 17 February 2009
Abstract
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.
- Type
- Research Article
- Information
- Copyright
- Copyright © Australian Mathematical Society 2006
References
- 1
- Cited by