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

Authors: Beong In Yun
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We introduce a generalized sigmoidal transformationwm(r;x) on a given interval [a,b] with a threshold at x=r∈(a,b). Using wm(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.


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## Body

## 1. Introduction

For a functionf 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 functionf having a jump-discontinuity ξ, 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 wm(r;x) with a threshold r=ξ. Using wm, we decompose the target function f into the left-hand part extension f~L and the right-hand part extension f~R as described in Section 3. Then we combine Fourier partial sums of f~L and f~R by the form of a weighted average, Fn, as given in (26) in Section 4. We prove the pointwise convergence of the presented approximation Fn to the discontinuous function f over the whole interval. Moreover, it is shown that the asymptotic version of Fn which is composed of uniform convergent partial sums will overcome the Gibbs phenomenon. This means that Fn 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[a,b] and some interior point a<r<b, referring to the literature [12], we introduce the real valued function(1)wmr;x=x-amx-am+r-a/b-rb-xm,a≤x≤bfor an integer m≥1. It was used for cumulative averaging method for piecewise polynomial interpolations in [12]. We call wm(x)=wm(r;x) a generalized sigmoidal transformation of order m with a threshold r.We can observe the basic properties ofwm as follows:(i)
The special case ofr=(a+b)/2, with a=0 and b=1, is(2)wmx=xmxm+1-xm,which is the same with the elementary sigmoidal transformation proposed in [13, 14].(ii)
Values ofwm(x) at the points x=a, r, and b are(3)wma=0,wmr=12,wmb=1independently of the parametersa, b, r, and m. In addition, wm(x) is strictly increasing on the interval [a,b] because the derivative of wm(x) with respect to x satisfies(4)wm′x=mb-ab-rmr-amb-xm-1x-am-1r-amb-xm+b-rmx-am2>0for alla<x<b.(iii)
Asymptotic behavior ofwm(x) near the end points a and b is(5)wmx=Ox-ar-am,x<r1+Ob-xb-rm,x>rasm goes to the infinity. Moreover, wm is sufficiently smooth over the interval (a,b); that is, wm(x)∈C∞(a,b).The generalized sigmoidal transformationwm 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 thatf is a piecewise smooth function containing a jump-discontinuity ξ in an interval [a,b]. We assume that the location of ξ or its accurate approximation is known and that the value f(ξ) is defined to be the average of the left- and right-hand limits of f at ξ; that is, (6)fξ=fξ-+fξ+2.On the other side, taking r=ξ in formula (1) or wm=wm(ξ;x), 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 Figure1:(7)f1x=15x-12-4,-π≤x<π3x+sin⁡3x,π3<x≤πwhich have a jump-discontinuity ξ=π/3:(8)f2x=12x+12,-π≤x≤πwhich is continuous on the interval [-π,π]. We notice that fi(-π)≠fi(π), i=1,2, and thus both f1 and f2 have jump-discontinuities at ±π when we extend these functions to the 2π periodic functions over the real line.Figure 1
Graphs of test functions.
(a)
f
1
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(b)
f
2
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)Let the piecewise smooth functionf(x), containing a jump-discontinuity ξ, be defined as(9)fx=fLx,a≤x<ξfRx,ξ<x≤b,where fL and fR are continuous on [a,ξ) and (ξ,b], respectively. We assume that the order m of wm=wm(ξ;x) is large enough throughout this paper. First, we define two quadratic polynomials QL and QR as(10)QLx=a-ξb-ξ2x-ξ2+ξsatisfying(11)QLξ=ξ,QLb=a,QRx=b-ξa-ξ2x-ξ2+ξsatisfying(12)QRξ=ξ,QRa=b.It should be noted that the left- and right-hand limits of QL and QR at ξ are(13)QLξ-=QLξ+=ξ-,QRξ-=QRξ+=ξ+.Then we construct extensions offL and fR onto the whole interval [a,b] as(14)f~Lx=1-wmxfx+wmxfQLx,(15)f~Rx=wmxfx+1-wmxfQRx,respectively, for a≤x≤b. It is seen that(16)f~Lξ=f~Rξ=fξ=fξ-+fξ+2as wm(ξ)=1/2.One can surmise that, for sufficiently largem, f~L has the effect of reflecting the left part fL of f on [a,ξ] into the opposite side [ξ,b]. So does f~R, symmetrically. In addition, these extended functions f~L and f~R defined in (14) and (15) have some particular properties as shown in the following lemmas.Lemma 1.
Letf be a piecewise smooth function on [a,b] with a jump-discontinuity ξ, and suppose that the order m of wm is fixed and finite. Then we have the one-sided limits of f~L and f~R as follows:(17)f~Lξ-=fξ-,f~Lξ+=fξ-+fξ+2,f~Rξ-=fξ-+fξ+2,f~Rξ+=fξ+.Furthermore,(18)f~La+=f~Lb-=fa,f~Ra+=f~Rb-=fb.Proof.
Sincelimx→ξwm(x)=1/2 and fQL(ξ-)=fQL(ξ+)=f(ξ-) for some m fixed, from (14) we have (19)f~Lξ-=12fξ-+fQLξ-=fξ-,f~Lξ+=12fξ++fξ-.By the same way, from (15) we have f~R(ξ-)=(1/2)f(ξ-)+f(ξ+) and f~R(ξ+)=f(ξ+).The equations in (18) directly result from the properties of QL, QR, and wm.Properties (17) in Lemma 1 imply that both f~L and f~R have the jump-discontinuity at ξ if the original function f has a jump-discontinuity such as f(ξ-)≠f(ξ+). The properties in (18) may resolve the troublesome problem in Fourier series approximation resulting from the mismatch at the end points.In Figure2, graphs of f~L and f~R for the test function f(x)=f1(x) with m=40, for example, illustrate the results in Lemma 1. Therein, thick lines indicate principal part fL(x) of the extended functions f~L in (a) and fR(x) of f~R in (b). Thin lines indicate reflected parts of fL(x) and fR(x) in (a) and (b), respectively, and dotted lines show the original graph of f(x).Figure 2
Graphs of the extended functionsf~L(x) and f~R(x) with m=40 for the test function f1(x).
(a)
f
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L
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(b)
f
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R
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)For sufficiently largem, however, we can see that the jump-discontinuities of f~L and f~R at ξ vanish as shown in the following lemma.Lemma 2.
For a functionf assumed in Lemma 1 both(20)f~Lξ+-f~Lξ-,f~Rξ+-f~Rξ-vanish as m goes to the infinity.Proof.
It follows thatwm(ξ-)=0 and wm(ξ+)=1 as m→∞. Thus from (14) and (15) we have f~L(ξ-)=f~L(ξ+)=f(ξ-) and f~R(ξ-)=f~R(ξ+)=f(ξ+). The proof is completed.Lemma2 indicates the asymptotic behavior of f~L and f~R below:(21)f~Lx~f~L,∞x≔fLx,a≤x≤ξfLQLx,ξ<x≤b,(22)f~Rx~f~R,∞x≔fRQRx,a≤x<ξfRx,ξ≤x≤bfor m large enough. It should be noted that(23)limx→ξf~L,∞x=fξ-,limx→ξf~R,∞x=fξ+.Thus, if we replace the values of fL and fR at ξ as fL(ξ)=f(ξ-) and fR(ξ)=f(ξ+), then both f~L,∞ and f~R,∞ are continuous on the whole interval [a,b].


## 4. Improving Fourier Partial Sum Approximation

In this section we assume that the piecewise smooth functionf is defined on [a,b]=[-π,π] with a jump-discontinuity -π<ξ<π. We consider Fourier series of f~L and f~R in the form of(24)Snf~Lx=∑k=-nnckLeikx,Snf~Rx=∑k=-nnckReikx,where ckL and ckR are Fourier coefficients defined as(25)ckL=∫-ππf~Lte-iktdt,ckR=∫-ππf~Rte-iktdt. Then we propose a weighted average of Snf~L and Snf~R as follows:(26)Fnx=1-wmxSnf~Lx+wmxSnf~Rx=∑k=-nnckL+ckR-ckLwmxeikxfor -π≤x≤π. It is noted that, like f~L and f~R, the weighted average Fn is discontinuous at ξ if f(ξ-)≠f(ξ+). Nevertheless, Fn has the meaningful convergence properties shown in the following theorem.Theorem 3.
Letf be a function assumed in Lemma 1 with [a,b]=[-π,π]. Then we have the following:(1)
For the orderm of wm fixed, the weighted average Fn(x) converges to(27)fx+Ohxm,hx=min⁡x+πξ+π,π-xπ-ξpointwise over the interval[-π,π], provided that the value of Fn at the jump-discontinuity ξ is defined as Fn(ξ)≔Fn(ξ-)+Fn(ξ+)/2.(2)
LetFn,∞ be a modified formula of Fn, in (26), obtained by replacing f~L and f~R by their asymptotic versions f~L,∞ and f~R,∞ defined in (21) and (22), respectively, with the assumptions fL(ξ)=f(ξ-) and fR(ξ)=f(ξ+). Then Fn,∞ converges to f pointwise over the interval [-π,π], getting out of the Gibbs phenomenon, as m,n→∞.Proof.
It is noted thatSnf~L and Snf~R, respectively, converge to f~L and f~R pointwise on the interval [-π,π] because f~L and f~R are both piecewise smooth [1].
Letx<ξ. Then from (26) and (5) we have (28)limn→∞⁡Fnx=limn→∞⁡Snf~Lx+Snf~Rx-Snf~Lxwmx=fx+Ox+πξ+πm. Similarly, for x>ξ, (29)limn→∞⁡Fnx=limn→∞⁡Snf~Rx+Snf~Lx-Snf~Rx1-wmx=fx+Oπ-xπ-ξm.For x=ξ, from definition (26) of Fn and results (17) in Lemma 1, we have the equations (30)limn→∞⁡Fnξ-=12f~Lξ-+f~Rξ-=3fξ-+fξ+4,limn→∞⁡Fnξ+=12f~Lξ++f~Rξ+=fξ-+3fξ+4.This implies that(31)limn→∞⁡Fnξ=limn→∞Fnξ-+Fnξ+2=fξ-+fξ+2=fξ.Therefore, the proof of the assertion thatFnconverges tofpointwise over the interval[-π,π]is completed.
For (2), it is clear from assertion (1) that Fn,∞ converges to f pointwise over the interval [-π,π] as m,n→∞. On the other hand, the definitions of f~L,∞ and f~R,∞ in (21) and (22), respectively, and the assumptions fL(ξ)=f(ξ-) and fR(ξ)=f(ξ+) imply that f~L,∞ and f~R,∞ are continuous at the original discontinuity ξ with f~L,∞(-π)=f~L,∞(π) and f~R,∞(-π)=f~R,∞(π). That is, f~L,∞ and f~R,∞ are free of jump-discontinuity at x=ξ,±π. Thus, the Fourier series Snf~L,∞ and Snf~R,∞ uniformly converge to f~L,∞ and f~R,∞, respectively. As a result, we can see that the weighted combination Fn,∞ of Snf~L,∞ and Snf~R,∞ will get out of the Gibbs phenomenon as m,n→∞. This completes the proof.Results of the approximations and errors ofFn(x) with n=8,16, for the test functions f1 and f2, are illustrated in Figures 3 and 4, respectively. Therein, we took the order of weight function as m=10n, for example. The results of Fn(x), indicated by the thick lines, are compared with those of the traditional Fourier partial sum approximation Snf(x) which are indicated by the thin lines. The figures show that the proposed approximation Fn(x) highly improves the Fourier partial sum approximation over the whole interval. In particular, the Gibbs phenomenon resulting from the interior jump-discontinuity or the mismatch at the end points has been resolved by Fn(x).Figure 3
Approximations of the weighted averageFn(x) in upper rows and the corresponding errors in lower rows for the test function f1(x). Thin lines indicate approximations and errors of the Fourier partial sum Snf(x).
(a)
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(b)
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16Figure 4
Approximations of the weighted averageFn(x) in upper rows and the corresponding errors in lower rows for the test function f2(x). Thin lines indicate approximations and errors of the Fourier partial sum Snf(x).
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(b)
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