Advances in Numerical Analysis

Volume 2016 (2016), Article ID 6758283, 10 pages

http://dx.doi.org/10.1155/2016/6758283

## Lebesgue Constant Using Sinc Points

^{1}German University in Cairo, New Cairo City 11835, Egypt^{2}Faculty of Science, Ain Shams University, Cairo 11566, Egypt^{3}University of Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany

Received 20 April 2016; Revised 20 July 2016; Accepted 18 August 2016

Academic Editor: William J. Layton

Copyright © 2016 Maha Youssef et al. 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

Lebesgue constant for Lagrange approximation at Sinc points will be examined. We introduce a new barycentric form for Lagrange approximation at Sinc points. Using Thiele’s algorithm we show that the Lebesgue constant grows logarithmically as the number of interpolation Sinc points increases. A comparison between the obtained upper bound of Lebesgue constant using Sinc points and other upper bounds for different set of points, like equidistant and Chebyshev points, is introduced.

#### 1. Introduction

The main point we wish to raise in this article is an effective measure of Lagrange approximation at a set of data defined by a conformal map generating the Sinc points [1].

It is well known that the Lebesgue constant associated with polynomial interpolation always increases at least logarithmically with the number of interpolation points [2]. It is also known that polynomial interpolation at equidistant points has an asymptotic exponential growth rate [3, 4]. This bad condition combined with Runge’s phenomena [5] makes polynomial interpolation at equidistant points sometimes useless for large number of interpolation points. In fact, since Faber and Bernstein [6, 7] established their negative results on interpolations, there were many articles demonstrating that with a careful choice of interpolation points an acceptable approximation can be gained if a special set of interpolation points is used [8]. These activities culminated in the famous conjecture by Erdős [9] who proved that there exists an optimal choice of points for Lagrange interpolation. This optimal set of points has so far never been found but some other sets of points might be a near optimal choice [10].

Most of the analytic and numerical estimations of Lebesgue constants discussed in literature are concentrating on Chebyshev points in connection with Lagrange interpolation [8]. There are also a few papers related to cardinal approximations based on Sinc functions [11]. We will compare our results in this paper with the derived formulas for Lagrange approximation using Chebyshev points and with Sinc approximations [11].

To find the upper bound of Lebesgue constant there are two approaches: the analytic approach and the numeric approach. The analytic techniques deliver a true upper bound of the Lebesgue constant using certain type of interpolation points [2, 8, 9, 11]. The numeric approach uses a table of Lebesgue constant values to find a least square fitting for these data. It is well known that a least square approximation of such data may fail to represent the asymptotic behavior. In our approach, we use the numeric values of the Lebesgue constant to reach the asymptotic behavior at infinity. For this reason, we use Sinc points approaching infinity as interpolation points in a barycentric form of Lagrange approximation in connection with Thiele’s algorithm.

This paper is organized as follows: Section 2 recalls the definition of Lagrange approximation at Sinc points and Lebesgue function/constant. Section 3 presents the barycentric formulas of Lagrange approximation. Section 4 discusses the use of Sinc points in a weighted barycentric formula. Section 5 contains the improved Lebesgue constant using Thiele’s algorithm. Section 6 collects some numeric experiments supporting our theoretical findings of the previous sections. Finally, the conclusions are given in Section 7.

#### 2. Definition of the Problem

Assume we have Sinc points on the interval . These Sinc points are defined as follows:where is a positive parameter denoting the step length on . These points are nonequidistant points created by a conformal mapping . The function maps the interval to the real line for which , . Lagrange approximation at the Sinc points is defined as [1] where are the function values at Sinc points and are the basis functions defined as where the function is given by

There are two properties to influence the performance of the approximation defined in (2), the conditioning and accuracy. The standard measures of these two factors are combined in the Lebesgue constant and Lebesgue function; for recent discussion see [15, 16].

The Lebesgue function associated with defined in (2) is given as

The maximum of this function is called Lebesgue constant [17]:

In this paper, we derive the asymptotic expression of Lebesgue constant asin which and .

#### 3. Barycentric Formula

The basis function defined in (3) can be written as [18] where is defined as in (4) and the sequence of weights is defined as This means that the Lagrange polynomial in (2) can be written asInterpolating the constant function to haveand dividing (10) by (11), it follows that

Rutishauser called (12) the second barycentric formula. This formula reduces the standard operations in Lagrange interpolation to only operations. It is also known that (12) is stable in the neighborhood of the interpolation points, which by some authors is called a well-conditioned approximation. In (12), the weights appear in the denominator exactly as in the numerator, except without the data factor . This means that any common factor in all weights may be canceled without affecting the value of .

From (2) and (12) we can derive thatThis means that the interpolation property is satisfied independently of the numbers as long as they are different from zero. Thus, the function interpolates the function between the given points for all choices of the weights , such that , [18]. If , then is the interpolating polynomial; for other choices of , is a rational function. In [19] it was shown that a rational interpolant represented as a quotient of polynomials is completely determined by its denominator. When the numerator does not have common factors with the denominator, the weights must alternate their sign for the rational function to have no poles in the interval or any another interval of interpolation [18].

Some explicit forms of have been derived for some specific types of interpolation points. For details on how to choose ; see [12, 18, 20, 21]. For the equidistant points on with spacing , the weights are [22]. For Chebyshev points of first kind, , the weights are given by . Another explicit form of weights can be given for Chebyshev points of second kind, , by , where [12]The common feature of all the weights is that they oscillate in sign and have the same magnitude , except for the first and last term [23].

The most simple weights, given in [12], are . In this case (14) is called Berrut’s first rational form [12]. For Berrut’s first rational form, the following has been proved [12] (Theorem 2.1).

Lemma 1 (see [12]). *Let , , be points; then *(1)*the rational function* *interpolates the function between the points , , and has no pole on the real line;*(2)* is a well-conditioned interpolation for a function between the points , .*

*Berrut’s first rational approximation has been examined by many authors, mainly to improve the use of equidistant points as interpolation points. The target of all the examinations is studying the Lebesgue constant for such approximation. For example, in [13], the following result for Lebesgue constant has been derived: A tighter upper bound of the Lebesgue constant has been obtained in [14]. The improved upper bound satisfies the following inequality:*

*The above results have been extended to more general nodes by Bos et al. in [24]. In their paper, Bos et al. introduced the notion and condition of well-spaced nodes guaranteeing the logarithmic behavior improving Lebesgue constant using Berrut’s first form. They demonstrated that any ordered well-spaced set of points created by a regular function can yield this logarithmic behavior. Note that, using the definition of regularity introduced in [24], we can prove that Sinc points are well-spaced nodes. This means that using Sinc points will guarantee the logarithmic behavior for Lebesgue constant.*

*In the next section we introduce binomial alternating weights that allow an improvement in the calculations of Lebesgue constant. We later compare the obtained Lebesgue constant for this approximation and the obtained Lebesgue constants mentioned above in (17) and (18) (see Figure 8).*

*4. Barycentric Formula at Sinc Points*

*4.1. Binomial Weights*

*Combining Berrut’s [12] and Schneider and Werner’s [18] result that the weights should be alternating and the concept of regularity offered in [24] will yield a smaller logarithmic level of Lebesgue constant. For this reason we introduce different weights . We shall show that using these weights in (14) will give a well-conditioned rational approximation with no poles on the real line . Also we gain a large improvement in the upper bound of Lebesgue constant. To this end, let us rename the Sinc points defined in (1) as Now define the weights . These weights are alternating and if summed up are zero.*

*Lemma 2. The polynomialis different from , for Sinc points and .*

*Proof. *The aim is to show that for all . To do so, we let We will study the asymptotes and the critical points of in the interval .

From (21) we haveThis means that has asymptotes at the Sinc points . To find the critical points of in (21), we follow the standard approach by equating the derivative to zero. The critical points in the interval exist at , where This critical point lies approximately at the middle of the interval . The relative local values of are given byThis means that has only one critical point in the interval . Moreover, has the following asymptotic behavior:where is a positive function in growing exponentially as . Since the structure of is much too complicated to be displayed, we present the plot of this function in Figure 1. Now we end up by the following two cases:(i)If and is even, then and and has only one minimal value. To check the sign of this minimal value, we first evaluate (24) at to get the minimal value equal to 6.35492, which is positive number. Also according to (25), for each , is exponentially increasing in . This means that is always positive in this case; see Figure 2(a) for demonstration.(ii) Similarly, if and is odd, then and and the sum has only one maximal value. At , this maximal value is equal to −6.35492 and has the asymptotic behavior in (25). This means that is always negative in this case; see Figure 2(a).This means that, for each , which completes the proof.