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

and Applied Analysis 3 enough throughout this paper. First, we define two quadratic polynomials QL and QR as QL (x) = a − ξ (b − ξ)2 (x − ξ) 2 + ξ (10) satisfying QL (ξ) = ξ, QL (b) = a, QR (x) = b − ξ (a − ξ)2 (x − ξ) 2 + ξ (11) satisfying QR (ξ) = ξ, QR (a) = b. (12) It should be noted that the leftand right-hand limits of QL and QR at ξ are QL (ξ−) = QL (ξ+) = ξ−, QR (ξ−) = QR (ξ+) = ξ+. (13) Thenwe construct extensions offL andfR onto the whole interval [a, b] as fL (x) = {1 − wm (x)} f (x) + wm (x) f (QL (x)) , (14) fR (x) = wm (x) f (x) + {1 − wm (x)} f (QR (x)) , (15) respectively, for a ≤ x ≤ b. It is seen that fL (ξ) = fR (ξ) = f (ξ) = (f (ξ−) + f (ξ+)) 2 (16) as wm(ξ) = 1/2. One can surmise that, for sufficiently large m, fL has the effect of reflecting the left part fL of f on [a, ξ] into the opposite side [ξ, b]. So does fR, symmetrically. In addition, these extended functions fL and fR defined in (14) and (15) have some particular properties as shown in the following lemmas. Lemma 1. Let f 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 offL and fR as follows: fL (ξ−) = f (ξ−) , fL (ξ+) = (f (ξ−) + f (ξ+)) 2 , fR (ξ−) = (f (ξ−) + f (ξ+)) 2 , fR (ξ+) = f (ξ+) . (17) Furthermore, fL (a+) = fL (b−) (= f (a)) , fR (a+) = fR (b−) (= f (b)) . (18) Proof. Since limx→ξ wm(x) = 1/2 and f(QL(ξ−)) = f(QL(ξ+)) = f(ξ−) for somem fixed, from (14) we have fL (ξ−) = 2 {f (ξ−) + f (QL (ξ−))} = f (ξ−) , fL (ξ+) = 2 {f (ξ+) + f (ξ−)} . (19) By the same way, from (15) we have fR(ξ−) = (1/2){f(ξ−) + f(ξ+)} and fR(ξ+) = f(ξ+). The equations in (18) directly result from the properties of QL, QR, and wm. Properties (17) in Lemma 1 imply that bothfL andfR 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 themismatch at the end points. In Figure 2, graphs of fL and fR 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 functionsfL in (a) andfR(x) offR in (b).Thin lines indicate reflected parts offL(x) andfR(x) in (a) and (b), respectively, and dotted lines show the original graph off(x). For sufficiently large m, however, we can see that the jump-discontinuities of fL and fR at ξ vanish as shown in the following lemma. Lemma 2. For a function f assumed in Lemma 1 both fL (ξ+) − fL (ξ−) , fR (ξ+) − fR (ξ−) (20) vanish asm goes to the infinity. Proof. It follows thatwm(ξ−) = 0 andwm(ξ+) = 1 asm → ∞. Thus from (14) and (15) we have fL(ξ−) = fL(ξ+) = f(ξ−) and fR(ξ−) = fR(ξ+) = f(ξ+). The proof is completed. Lemma 2 indicates the asymptotic behavior of fL and fR below: fL (x) ∼ fL,∞ (x) fl {{{ fL (x) , a ≤ x ≤ ξ fL (QL (x)) , ξ < x ≤ b, (21) fR (x) ∼ fR,∞ (x) fl {{{ fR (QR (x)) , a ≤ x < ξ fR (x) , ξ ≤ x ≤ b (22) form large enough. It should be noted that lim x→ξ fL,∞ (x) = f (ξ−) , lim x→ξ fR,∞ (x) = f (ξ+) . (23) Thus, if we replace the values of fL and fR at ξ as fL(ξ) = f(ξ−) and fR(ξ) = f(ξ+), then both fL,∞ and fR,∞ are continuous on the whole interval [a, b]. 4 Abstract and Applied Analysis


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 socalled 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][4][5], the inverse reconstruction [6,7], and the adaptive filtering method [8][9][10][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 jumpdiscontinuity , 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 f and the right-hand part extension f as described in Section 3. Then we combine Fourier partial sums of f and f 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.

A Generalized Sigmoidal Transformation
For a given interval [, ] and some interior point  <  < , referring to the literature [12], we introduce the real valued function for an integer  ≥ 1.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  = ( + )/2, with  = 0 and  = 1, 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.

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  = /3: which is continuous on the interval [−, ].We notice that   (−) ̸ =   (),  = 1, 2, and thus both  1 and  2 have jumpdiscontinuities at ± when we extend these functions to the 2 periodic functions over the real line.
Let the piecewise smooth function (), containing a jump-discontinuity , be defined as where   and   are continuous on [, ) and (, ], respectively.We assume that the order  of   =   (; ) is large enough throughout this paper.First, we define two quadratic polynomials   and   as satisfying satisfying It should be noted that the left-and right-hand limits of   and   at  are Then we construct extensions of   and   onto the whole interval [, ] as respectively, for  ≤  ≤ .It is seen that as   () = 1/2.One can surmise that, for sufficiently large , f has the effect of reflecting the left part   of  on [, ] into the opposite side [, ].So does f , symmetrically.In addition, these extended functions f and f defined in (14) and (15) have some particular properties as shown in the following lemmas.For sufficiently large , however, we can see that the jump-discontinuities of f and f at  vanish as shown in the following lemma.

Improving Fourier Partial Sum Approximation
In this section we assume that the piecewise smooth function  is defined on [, ] = [−, ] with a jump-discontinuity − <  < .We consider Fourier series of f and f in the form of where    and    are Fourier coefficients defined as Then we propose a weighted average of   f and   f as follows: for − ≤  ≤ .It is noted that, like f and f , the weighted average   is discontinuous at  if (−) ̸ = (+).Nevertheless,   has the meaningful convergence properties shown in the following theorem.
(2) Let  ,∞ be a modified formula of   , in (26), obtained by replacing f and f by their asymptotic versions f,∞ and f,∞ defined in ( 21) and ( 22 Therefore, the proof of the assertion that   converges to  pointwise over the interval [−, ] is completed.For (2), it is clear from assertion (1) that  ,∞ converges to  pointwise over the interval [−, ] as ,  → ∞.On the other hand, the definitions of f,∞ and f,∞ in (21) and ( 22), respectively, and the assumptions   () = (−) and   () = (+) imply that f,∞ and f,∞ are continuous at the original discontinuity  with f,∞ (−) = f,∞ () and f,∞ (−) = f,∞ ().That is, f,∞ and f,∞ are free of jumpdiscontinuity at  = , ±.Thus, the Fourier series   f,∞ and   f,∞ uniformly converge to f,∞ and f,∞ , respectively.As a result, we can see that the weighted combination  ,∞ of   f,∞ and   f,∞ will get out of the Gibbs phenomenon as ,  → ∞.This completes the proof.
Results of the approximations and errors of   () with  = 8, 16, for the test functions  1 and  2 , are illustrated in Figures 3 and 4, respectively.Therein, we took the order of weight function as  = 10, for example.The results  of   (), indicated by the thick lines, are compared with those of the traditional Fourier partial sum approximation   () which are indicated by the thin lines.The figures show that the proposed approximation   () 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   ().
we replace the values of   and   at  as   () = (−) and   () = (+), then both f,∞ and f,∞ are continuous on the whole interval [, ].

Figure 2 :
Figure 2: Graphs of the extended functions f () and f () with  = 40 for the test function  1 ().

Figure 3 :
Figure 3: Approximations of the weighted average   () in upper rows and the corresponding errors in lower rows for the test function  1 ().Thin lines indicate approximations and errors of the Fourier partial sum   ().

Figure 4 :
Figure 4: Approximations of the weighted average   () in upper rows and the corresponding errors in lower rows for the test function  2 ().Thin lines indicate approximations and errors of the Fourier partial sum   ().