Abstract and Applied Analysis

Volume 2017, Article ID 1364914, 7 pages

https://doi.org/10.1155/2017/1364914

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

Department of Mathematics, Kunsan National University, Gunsan, Republic of Korea

Correspondence should be addressed to Beong In Yun; moc.liamg@nuyllluap

Received 1 September 2017; Revised 16 October 2017; Accepted 19 October 2017; Published 22 November 2017

Academic Editor: Roberto Barrio

Copyright © 2017 Beong In Yun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a generalized sigmoidal transformation on a given interval with a threshold at . Using , 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.

#### 1. Introduction

For a function having a jump-discontinuity, every traditional spectral partial sum approximation will not converge uniformly on any interval containing the discontinuity. This deficiency of the spectral approximation results in the so-called Gibbs phenomenon which shows nonvanishing spikes near the discontinuity [1, 2]. There are lots of methods to overcome the problem such as the Fourier-Gegenbauer method [3–5], the inverse reconstruction [6, 7], and the adaptive filtering method [8–11]. But most existing methods need a large number of terms to support high accuracy.

In this work, focusing on the Fourier partial sum approximation for a piecewise smooth function having a jump-discontinuity , we aim to develop a constructive approximation procedure which is available for eliminating the Gibbs phenomenon near the discontinuity. First, in the following section, we introduce the so-called generalized sigmoidal transformation with a threshold . Using , we decompose the target function into the left-hand part extension and the right-hand part extension as described in Section 3. Then we combine Fourier partial sums of and by the form of a weighted average, , as given in (26) in Section 4. We prove the pointwise convergence of the presented approximation to the discontinuous function over the whole interval. Moreover, it is shown that the asymptotic version of which is composed of uniform convergent partial sums will overcome the Gibbs phenomenon. This means that can sufficiently resolve the problem of inevitable wiggles of the traditional Fourier partial sum approximation near the jump-discontinuity. In addition, numerical results for some examples show the availability of the presented method.

#### 2. A Generalized Sigmoidal Transformation

For a given interval and some interior point , referring to the literature [12], we introduce the real valued functionfor an integer . It was used for cumulative averaging method for piecewise polynomial interpolations in [12]. We call a generalized sigmoidal transformation of order with a threshold .

We can observe the basic properties of as follows:(i)The special case of , with and , is which is the same with the elementary sigmoidal transformation proposed in [13, 14].(ii)Values of at the points , , and are independently of the parameters , , , and . In addition, is strictly increasing on the interval because the derivative of with respect to satisfies for all .(iii)Asymptotic behavior of near the end points and is as goes to the infinity. Moreover, is sufficiently smooth over the interval ; that is, .

The generalized sigmoidal transformation plays an important role in developing a new approximation method as a weight function in this work.

#### 3. Decomposition of a Discontinuous Function

From now on we suppose that is a piecewise smooth function containing a jump-discontinuity in an interval . We assume that the location of or its accurate approximation is known and that the value is defined to be the average of the left- and right-hand limits of at ; that is, On the other side, taking in formula (1) or , we will use it as a weight function for the proposed approximation method in this work.

We choose the test functions below whose graphs are given in Figure 1:which have a jump-discontinuity :which is continuous on the interval . We notice that , , and thus both and have jump-discontinuities at when we extend these functions to the periodic functions over the real line.