Lebesgue functions and Lebesgue constants in polynomial interpolation

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.

Least-squares collocation for linear higher-index differential–algebraic equations

“…there is a set of interpolation nodes τ * i which minimizes the corresponding Lebesgue constant Λ * M . This constant is only slightly smaller than Λ C M [34]. For computing the Gauss-Lobatto nodes and weights, the methods of [41] (using the Newton method) as well as [21] (a variant of the method in [26]) have been implemented.…”

Bayesian Model Calibration with Interpolating Polynomials Based on Adaptively Weighted Leja Nodes

“…is the operator norm of L N induced by the norm • ∞ discussed above. Λ N is called the Lebesgue constant [16]. The inequality holds for any polynomial ϕ N , so an immediate result is the Lebesgue inequality:…”

“…For example, Chebyshev nodes (i.e. the nodes from the Clenshaw-Curtis quadrature rule) have Λ N = O(logN) [16]. Moreover, we already stated that the Chebyshev nodes are nested such that the nodes for N = 2 l +1 (for integer l) are contained in the nodes for N = 2 l+1 +1.…”

“…As extensive theory about interpolation polynomials exists (e.g. [16]), we can prove convergence of the estimated posterior with mild assumptions on the likelihood. This extends previous work [4,24], in which the likelihood is assumed to be Gaussian.…”

Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations—Part 1: basics and ansatz function choices

“…6 For each M, there is a set of interpolation nodes τ * i that minimizes the corresponding Lebesgue constant Λ * M . This constant is only slightly smaller than Λ C M [30]. For each method, the absolute error of the nodes (A), the absolute error of the weights (B), the maximum componentwise relative error of nodes (C) and weights (D) are shown.…”