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.
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Let the piecewise smooth function , containing a jump-discontinuity , be defined aswhere 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 assatisfyingsatisfyingIt should be noted that the left- and right-hand limits of and at are
Then we construct extensions of and onto the whole interval asrespectively, for . It is seen thatas .
One can surmise that, for sufficiently large , has the effect of reflecting the left part of on into the opposite side . So does , symmetrically. In addition, these extended functions and defined in (14) and (15) have some particular properties as shown in the following lemmas.
Lemma 1. Let be a piecewise smooth function on with a jump-discontinuity , and suppose that the order of is fixed and finite. Then we have the one-sided limits of and as follows:Furthermore,
Proof. Since and for some fixed, from (14) we have By the same way, from (15) we have and The equations in (18) directly result from the properties of , , and .
Properties (17) in Lemma 1 imply that both and have the jump-discontinuity at if the original function has a jump-discontinuity such as . The properties in (18) may resolve the troublesome problem in Fourier series approximation resulting from the mismatch at the end points.
In Figure 2, graphs of and for the test function with , for example, illustrate the results in Lemma 1. Therein, thick lines indicate principal part of the extended functions in (a) and of in (b). Thin lines indicate reflected parts of and in (a) and (b), respectively, and dotted lines show the original graph of .
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For sufficiently large , however, we can see that the jump-discontinuities of and at vanish as shown in the following lemma.
Lemma 2. For a function assumed in Lemma 1 bothvanish as goes to the infinity.
Proof. It follows that and as . Thus from (14) and (15) we have and . The proof is completed.
Lemma 2 indicates the asymptotic behavior of and below:for large enough. It should be noted thatThus, if we replace the values of and at as and , then both and are continuous on the whole interval .
4. 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 and in the form ofwhere and are Fourier coefficients defined as Then we propose a weighted average of and as follows:for . It is noted that, like and , the weighted average is discontinuous at if . Nevertheless, has the meaningful convergence properties shown in the following theorem.
Theorem 3. Let be a function assumed in Lemma 1 with . Then we have the following:(1)For the order of fixed, the weighted average converges to pointwise over the interval , provided that the value of at the jump-discontinuity is defined as .(2)Let be a modified formula of , in (26), obtained by replacing and by their asymptotic versions and defined in (21) and (22), respectively, with the assumptions and . Then converges to pointwise over the interval , getting out of the Gibbs phenomenon, as .
Proof. It is noted that and , respectively, converge to and pointwise on the interval because and are both piecewise smooth [1].
Let. Then from (26) and (5) we have Similarly, for , For , from definition (26) of and results (17) in Lemma 1, we have the equations This implies thatTherefore, the proof of the assertion thatconverges topointwise over the intervalis completed.
For (), it is clear from assertion () that converges to pointwise over the interval as . On the other hand, the definitions of and in (21) and (22), respectively, and the assumptions and imply that and are continuous at the original discontinuity with and . That is, and are free of jump-discontinuity at . Thus, the Fourier series and uniformly converge to and , respectively. As a result, we can see that the weighted combination of and will get out of the Gibbs phenomenon as . This completes the proof.
Results of the approximations and errors of with , for the test functions and , are illustrated in Figures 3 and 4, respectively. Therein, we took the order of weight function as , 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 .
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Conflicts of Interest
The author declares that he has no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2017R1A2B4007682).