On Some Extremal Properties of Lagrange Interpolatory Polynomials Original Research Article

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths. In particular, we will show thatinfLn∈ℓnsupp∈Pn+1 ∥p−Lnp∥C[−1,1]=12n , where Ln is a set of the linear operators with finite rank n+1 defined on C−1,1], and where Pn+1 denotes the set of polynomials p=∑i=0n+1aixi of degree⩽n+1 such that ∣an+1∣⩽1. The infimum is achieved for Lagrange interpolatory polynomial for nodes 2k+12(n+1), k=0,…,n.