Error Bounds for Lagrange Interpolation Original Research Article

In this paper we study the quantities [formula] which define error bounds for the approximation of functions ƒ ∈ Wm∞[a, b] by the interpolating Lagrange polynomials lm−1, Δ(f) of degree m − 1, constructed on the given mesh of interpolating nodes Δ = Δm = {a ≤ t1 < ··· < tm ≤ b}. Set [formula] It is clear that Lm, k(Δ, x) ≥ 1m!|ω(k)Δ(x)|, Lm, k(Δ) ≥ 1m!∥ω(k)Δ(·)∥. Our main result is THEOREM 1. For all m and k (0 ≤ k ≤ m − 1), and for any mesh Δ of the interpolating nodes {ti}m1Lm, k(Δ) = 1m!∥ω(k)Δ(·)∥.