Lebesgue functions and Lebesgue constants in polynomial interpolation
- Bayram Ali Ibrahimoglu^{1}Email author
https://doi.org/10.1186/s13660-016-1030-3
© Ibrahimoglu 2016
Received: 7 July 2015
Accepted: 20 February 2016
Published: 12 March 2016
Abstract
The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant is computed by using the Lagrange basis, then the Lebesgue constant also expresses the conditioning of the interpolation problem. In addition, many publications have been devoted to the search for optimal interpolation points in the sense that these points lead to a minimal Lebesgue constant for the interpolation problems on the interval \([-1, 1]\).
In Section 1 we introduce the univariate polynomial interpolation problem, for which we give two useful error formulas. The conditioning of polynomial interpolation is discussed in Section 2. A review of some results for the Lebesgue constants and the behavior of the Lebesgue functions in view of the optimal interpolation points is given in Section 3.
Keywords
1 Introduction
2 Lebesgue function and constant
2.1 Definition and properties
The following present some basic properties of Lebesgue functions for Lagrange interpolation (see, e.g., [2, 3]):
(a) For any set X, with \(n\ge2 \), \(L_{n} (X;x ) \) is a piecewise polynomial satisfying \(L_{n} (X;x )\ge1 \) with equality only at the interpolation points \(x_{j}\), \(j=0, \ldots, n \).
(b) On each subinterval \((x_{j-1}, x_{j} ) \) for \(1\le j\le n \), \(L_{n} (X;x ) \) has precisely one local maximum, which is denoted by \(\lambda_{j} (X ) \). If the endpoints −1 and +1 are not interpolation points, that is, \(-1< x_{0} \) and \(x_{n} <1 \), then there are two other subintervals and, thus, two other local maxima that are at −1, and +1. We denote the latter two local maxima by \(\lambda_{0} (X ) \) and \(\lambda_{n+1} (X ) \).
(c) The greatest and the smallest local maxima of \(L_{n} (X;x ) \) are denoted correspondingly by \({{\mathcal {M}}}_{n} (X )\) and \(m_{n} (X ) \); we denote by \(\delta _{n} (X ) \) the maximum deviation among the local maxima \(\delta_{n} (X )={\mathcal {M}}_{n} (X )-m_{n} (X ) \). We also denote the position of the Lebesgue constant (by taking one of the greatest local maxima) by \(x^{*} (X )\) for the set of interpolation points X.
(d) The equality \(L_{n} (X;x )=L_{n} (X;-x )\), \(x\in [-1, 1 ]\), holds if and only if \(x_{n-j} =-x_{j}\), \(j=0, \ldots, n\).
We use these properties in the sequel.
2.2 Importance of Lebesgue constants
This indicates that if we are able to choose interpolation points such that \(\Lambda_{n} \) is small, then we can find the Lagrange interpolant that is less sensitive to errors in the function values. For this reason, numerical interpolation in floating-point arithmetic will generally be useless, even for smooth functions f, whenever the Lebesgue constant \(\Lambda_{n} \) is larger than the inverse of the machine precision, which is typically about 10^{16}.
3 Some specific sets of interpolation points
This section gives a summary of some results for particular sets of interpolation points for which the behavior of the Lebesgue function has been well investigated.
3.1 Equidistant nodes
There are many studies on the behavior of the Lebesgue function corresponding to the set of equidistant points although this set is a bad choice for polynomial interpolation owing to the Runge phenomenon.
From the Lebesgue inequality (6) we know that equidistant points with this very fast growth of the Lebesgue constant give very poor approximations as n increases. Indeed, numerical experiments show that for degree \(n\ge65 \), the Lebesgue constant \(\Lambda_{n} (E ) \) reaches the inverse of the machine precision.
3.2 Chebyshev nodes of the first kind
As Figure 1 (center) suggests, the local maxima of \(L_{n} (T;x ) \) are decreasing strictly from the outside toward the midpoint of the interval \([-1, 1]\), which was proven in [13]. The figure also shows that the location of the Lebesgue constant occurs at ±1, that is, \(x^{*} (T )=\pm1 \) [15, 16].
3.3 Extended Chebyshev nodes
3.4 Chebyshev extrema
As was proven in [13], the local maxima of \(L_{n} (\bar{U};x ) \) increase strictly monotonically from the outside toward the midpoint of the interval \([-1, 1]\). This behavior suggests that the Lebesgue function \(L_{n} (\bar{U};x ) \) achieves its maximum value on the subinterval \((x_{{n/ 2} }, x_{{ (n+2 )/ 2} } ) \) (or its mirror) for even degrees and on the subinterval \((x_{{ (n-1 )/ 2} }, x_{{ (n+1 )/ 2} } ) \) for odd degrees.
Numerical observation indicates that the location of the Lebesgue constant occurs at \(x^{*} (\bar{U} )\approx\frac{\pi}{2n} \) (or its mirror) for (large) even degrees and at \(x^{*} (\bar {U} )=0 \) for odd degrees.
3.5 Chebyshev nodes of the second kind
Note that these interpolation points can be obtained from the zeros of the polynomial \((1-x^{2} )T'_{n+2} (x ) \) by deleting the zeros ±1. Thus, it follows that the Lebesgue constants are sensitive to the deletion of the endpoints.
3.6 Fekete nodes
3.7 Optimal nodes
These conjectures were proven by Kilgore [28, 29] and by de Boor and Pinkus [30]. They showed that for degree n, the optimal canonical interpolation set is unique, symmetric, and that its Lebesgue function must necessarily equioscillate. By using these basic properties of the optimal nodes a numerical procedure based on a nonlinear Remez search and exchange algorithm is given to compute the optimal nodes for polynomial interpolation on \([-1, 1] \) [31]. Moreover, many authors (see, e.g., [22, 32]) have investigated (near) optimal point sets (in different norms) defined by the solutions of certain optimization problems.
The values of the maximum deviations and Lebesgue constants for sets \(\pmb{\breve{T} }\) , \(\pmb{\hat{T} }\) , and \(\pmb{X^{*}}\)
n | set \(\boldsymbol {\breve{T}}\) | set \(\boldsymbol {\hat{T}}\) | set \(\boldsymbol {X^{*}}\) | ||
---|---|---|---|---|---|
\(\boldsymbol {\delta _{n} (\breve{T})}\) | \(\boldsymbol {\Lambda _{n} (\breve{T})}\) | \(\boldsymbol {\delta _{n} (\hat{T})}\) | \(\boldsymbol {\Lambda _{n} (\hat{T})}\) | \(\boldsymbol {\Lambda _{n} (X^{*} )}\) | |
10 | 0.050 781 | 2.056 087 | 0.019 471 | 2.068 744 | 2.051 706 |
20 | 0.056 995 | 2.463 129 | 0.019 340 | 2.479 193 | 2.460 788 |
40 | 0.061 827 | 2.887 067 | 0.018 952 | 2.904 441 | 2.885 809 |
4 Concluding remarks
In this paper we work with the interval \([- 1, 1] \) although all results on polynomial interpolation may be applied to any finite interval by a linear change of variable.
In view of the optimal interpolation points for the univariate polynomial interpolation, to our knowledge, both sets T̆ and T̂, with the position of the points given in explicit form, are the best nodal sets in the literature. Based on the Bernstein-Erdös conjecture, the nodal set T̂ is superior than the nodal set T̆ because of its smaller maximum deviation. When considering the optimality of a nodal set from its Lebesgue constant, the set T̆ is better.
In the multivariate case, the problem of optimal or near optimal interpolation is much more difficult. The minimal growth of the Lebesgue constant is different for different bivariate domains. For instance, on the square, the minimal order of growth is \(O(\ln^{2}(n+1))\), and this order is achieved for the configurations of interpolation points given in [36] and [37]. On the disk, the minimal order of growth is quite different, namely \(O(\sqrt{n+1})\), as proved in [38]. No configurations of interpolation points obeying this order of growth are known. On the simplex, the minimal order of growth is not even known.
Declarations
Acknowledgements
The author expresses his sincere thanks to Professor Annie Cuyt for her kind suggestions. This work was partially supported by the Scientific and Technical Research Council of Turkey (TUBITAK) under Grant No. 2214/B.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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