Estimating the Lebesgue constant for the Chebyshev distribution of nodes

Abstract

In this paper an approach to estimation of the Lebesgue constant for the Lagrange interpolation process with nodes in the zeros of Chebyshev polynomials of the first kind is done. Two-sided estimation of this constant is carried out by using the logarithmic derivative of the Euler gamma function and of the Riemann zeta function. The choice of interpolation nodes is due to the fact that with a fixed number of Chebyshev nodes, the Lebesgue constant tends to its minimum value, thus reducing the error of algebraic interpolation and providing less sensitivity to rounding errors. The expressions for the upper and the lower bounds of this constant are represented as finite sums of an asymptotic alternating series. Based on the expressions obtained, these boundaries are calculated depending on the number of nodes of the interpolation process. The error of each of the boundaries’ value is estimated based on the first discarded term in the corresponding asymptotic series. The results of the calculations are presented in tables showing deviations of the Lebesgue constant from its lower and upper estimated bounds. Dependence of the values’ errors on the number of Chebyshev nodes is depicted in these tables as well. It is numerically shown that with an increase in the number of these nodes, the estimation boundaries rapidly get close to each other. The presented results can be used in the theory of interpolation to estimate the norm of the operator matching a function to its interpolation polynomial and to estimate a deviation of the constructed perturbed polynomial from the unperturbed one.