Lebesgue Constants and Optimal Node Systems via Symbolic Computations

Polynomial interpolation is a classical method to approximatecontinuous functions by polynomials. To measure the correctness of theapproximation, 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 asymptoticbehavior 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 minimalLebesgue constants symbolically on $[-1,1]$ and we are also looking for thecharacterization of the those node systems which realize theminimal Lebesgue constants. We exploited the equioscillation property of the Lebesgue function \cite{4} andused quantifier elimination and Groebner Basis as tools \cite{1,2}. Most of the computation is done in Mathematica \cite{8}.