Polynomial Interpolation In Points Equidistributed On The Unit Circle

Polynomial Interpolation in Points Equidistributed on the Unit Circle

A Stochastic Treatment of Some Classical Interpolation Problems

Polynomial Interpolation in Points Equidistributed on the Unit Circle

Let Ln(f; z) be the polynomial of degree at most n-1 found by interpolation in the distinct points znk = ei nk, k = 1 ..., n, to a function f given on z = 1. It is known that a nece sary and sufficient condition that nlim Ln(f; z) = f(z), z 1, for all f analytic n z 1, is that nk be equidistributed on 0, 2 . In nonanalytic cases, convergence has been established when znk is an n-th root of unity, b t the behavior of Ln with other spacings of the interpolation points is not clear. It is here proved that if nk, k = 1 ..., n, are independent random variables each with a uniform probability distribution and if f satisfies certain mild smoothness restrictions on z = 1, then where Ln is found by interpolation to f in the random points znk = ei nk. A simple example is constructed involving an equidistributed sample seq ence nk for w ich Ln(f; z) diverges to infinity at each point z, z