The ANZIAM Journal

Table of Contents - 2008 - Volume 49, Issue 03  

Research Article

SIGMOIDAL-TYPE SERIES EXPANSION

BEONG IN YUNa1

a1 School of Mathematics, Informatics and Statistics, Kunsan National University, Kunsan, 573-701, South Korea (email: biyun@kunsan.ac.kr)

Abstract

In this paper we introduce a set of orthonormal functions, $\{\phi _n^{[r]}\}_{n=1}^{\infty }$, where xs03D5n[r] is composed of a sine function and a sigmoidal transformation γr of order r>0. Based on the proposed functions xs03D5n[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 xs211D and then, by reconstructing the existing sigmoidal cosine functions ψn[r] and the presented functions xs03D5n[r], we develop two kinds of 2-periodic series expansions on xs211D. Superiority of the presented sigmoidal-type series in approximating a function by the partial sum is demonstrated by numerical examples.

(Received August 03 2006)

(Revised January 10 2008)

2000 Mathematics subject classification

  • primary 41A58; secondary 42A20

Keywords and phrases

  • Fourier series;
  • sigmoidal transformation;
  • sigmoidal series