Dolomites Research Notes on Approximation

Suppose that K ⊂ C is compact and that z0 ∈ C\K is an external point. An optimal prediction measure for regression by polynomials of degree at most n, is one for which the variance of the prediction at z0 is as small as possible. Hoel and Levine ([4]) have considered the case of K = [−1,1] and z0 = x0 ∈ R\[−1, 1], characterizing the optimal measures. More recently, [2] has given the equivalence of the optimal prediction problem with that of finding polynomials of extremal growth. They also study in detail the case of K = [−1, 1] and z0 = ia ∈ iR, purely imaginary. In this work we find, for these two cases, the limits of the optimal prediction measures as n→∞and show that they are the push-forwards via conformal mapping of the Poisson kernel measure for the disk. Moreover, in the case of z0 = ia ∈ iR, we show that the optimal prediction measure of degree n is actually the Gauss-Lobatto quadrature formula for this limiting push-forward measure.