## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2017 (2017), Article ID 1364914, 7 pages.

### Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function

#### Abstract

We introduce a generalized sigmoidal transformation ${w}_{m}(r;x)$ on a given interval $[a,b]$ with a threshold at $x=r\in (a,b)$. Using ${w}_{m}(r;x)$, we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2017 (2017), Article ID 1364914, 7 pages.

**Dates**

Received: 1 September 2017

Revised: 16 October 2017

Accepted: 19 October 2017

First available in Project Euclid: 14 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1513220444

**Digital Object Identifier**

doi:10.1155/2017/1364914

**Mathematical Reviews number (MathSciNet)**

MR3731731

**Zentralblatt MATH identifier**

06929543

#### Citation

Yun, Beong In. Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function. Abstr. Appl. Anal. 2017 (2017), Article ID 1364914, 7 pages. doi:10.1155/2017/1364914. https://projecteuclid.org/euclid.aaa/1513220444