Lebesgue Constants and Optimal Node Systems via Symbolic Computations

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 = {x1 < . . . < xn+1} (xj ∈ [a, b]), the Lebesgue function λn(x) is the sum of the modulus of the Lagrange basis polynomials built on X. The Lebesgue constant Λn assigned to the function λn(x) is its maximum over [a, b]. The Lebesgue constant bounds the interpolation error, i.e., the interpolation polynomial is at most (1 + Λn) times worse then the best approximation. The minimum of the Λn’s for fixed n and interval [a, b] is called the optimal Lebesgue constant Λ ∗ n. For specific interpolation node systems such as the equidistant system, numerical results for the Lebesgue constants Λn and their asymptotic behavior are known [3, 7]. However, to give explicit symbolic expression for the minimal Lebesgue constant Λn is computationally difficult. In this work, motivated by Rack [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 [4] and used quantifier elimination and Groebner Basis as tools [1, 2]. Most of the computation is done in Mathematica [8]. Acknowledgement. The research of the author was partially supported by the HSRF (OTKA), grant number K83219.