Lebesgue Constants and Optimal Node Systems via Symbolic Computations

1 pagesPublished: June 19, 2013

Abstract

Polynomial interpolation is a classical method to approximate
continuous functions by polynomials. To measure the correctness of the
approximation, Lebesgue constants are introduced. For a given node system $X^{(n+1)}=\{x_1<\ldots<x_{n+1}\}\, (x_j\in [a,b])$, the Lebesgue function $\lambda_n(x)$ is the sum of the modulus of the Lagrange basis polynomials built on $X^{(n+1)}$. The Lebesgue constant $\Lambda_n$ assigned to the function $\lambda_n(x)$ is its maximum over $[a,b]$. The Lebesgue constant bounds the interpolation error, i.e., the interpolation polynomial is at most $(1+\Lambda_n)$ times worse then the best approximation.
The minimum of the $\Lambda_n$'s for fixed $n$ and interval $[a,b]$ is called the optimal Lebesgue constant $\Lambda_n^*$.
For specific interpolation node systems such as the equidistant system, numerical results for the Lebesgue constants $\Lambda_n$ and their asymptotic
behavior are known \cite{3,7}. However, to give explicit symbolic expression for the minimal Lebesgue constant $\Lambda_n^*$ is computationally difficult. In this work, motivated by Rack \cite{5,6}, we are interested for expressing the minimal
Lebesgue constants symbolically on $[-1,1]$ and we are also looking for the
characterization of the those node systems which realize the
minimal Lebesgue constants. We exploited the equioscillation property of the Lebesgue function \cite{4} and
used quantifier elimination and Groebner Basis as tools \cite{1,2}. Most of the computation is done in Mathematica \cite{8}.

Keyphrases: algebraic number, Groebner basis, Lebesgue constant, polynomial interpolation, quantifier elimination, resultant

In: Laura Kovács and Temur Kutsia (editors). SCSS 2013. 5th International Symposium on Symbolic Computation in Software Science, vol 15, pages 125--125