## Abstract and Applied Analysis

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

Beong In Yun

#### 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

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