Balayage and Convergence of Rational Interpolants

Abstract

We investigate the following problem: For which open simply connected domains do there exist interpolation schemes (a set of interpolation points) such that for any analytic function defined in the domain the corresponding interpolating polynomials converge to the function when the degree of the polynomials tends to infinity? We also study similar problems for rational interpolants. These problems are connected to the balayage (sweeping out) problems of measures. Academic Press

Rational and Polynomial Interpolation of Analytic Functions with Restricted Growth

“…The crux is, as we saw in the reasoning above, that for non analytic boundaries we have to let the interpolation points approach the boundary, while for analytic boundaries, the interpolation points can be separated from the boundary. When we let the interpolation points approach the boundary, it is natural to expect (and formally proven in [3,Theorem 5]) that we cannot obtain convergence for any analytic function on D. Analytic functions can behave very wildly near the boundary and we may get ''bad information'' by interpolating at such points.…”

“…This type of problem has been studied by Walsh (see [8]) and Bagby (see [4]) and more recently by Ambroladze and Wallin in the papers [1], [2] and [3]. If we consider interpolating polynomials (which corresponds to the case where all poles are at infinity) on a bounded simply connected domain D … C, and apply results from [1] and [3] we have the following.…”

“…If we consider interpolating polynomials (which corresponds to the case where all poles are at infinity) on a bounded simply connected domain D … C, and apply results from [1] and [3] we have the following.…”