On a Weighted Interpolation of Functions with Circular Majorant

Denote by Ln the projection operator obtained by applying the Lagrange interpolation method, weighted by (1 − x), at the zeros of the Chebyshev polynomial of the second kind of degree n + 1. The norm ‖Ln‖ = max ‖f‖∞≤1 ‖Lnf‖∞, where ‖ · ‖∞ denotes the supremum norm on [−1, 1], is known to be asymptotically the same as the minimum possible norm over all choices of interpolation nodes for unweighted Lagrange interpolation. Because the projection forces the interpolating function to vanish at ±1, it is appropriate to consider a modified projection norm ‖Ln‖ψ = max |f(x)|≤ψ(x) ‖Lnf‖∞, where ψ ∈ C[−1, 1] is a given function (a curved majorant) that satisfies 0 ≤ ψ(x) ≤ 1 and ψ(±1) = 0. In this paper the asymptotic behaviour of the modified projection norm is studied in the case when ψ(x) is the circular majorant w(x) = (1 − x). In particular, it is shown that asymptotically ‖Ln‖w is smaller than ‖Ln‖ by the quantity 2π−1(1− log 2).