The Convergence of Polynomial Interpolation and Runge Phenomenon

Abstract

One of the main questions is whether or not a sequence of polynomials pn(x) that interpolate a continuous function f at n + 1 equally spaced points tends to f in the sup-norm? The answer is "no" in some cases. The main fact is that interpolant polynomials pn(x) of a function f converge at a rate determined by the smoothness of f: the pn(x) converge rapidly to the function f if it is k-times differentiable and converges exponentially if f is analytic. The polynomial interpolation depends on n but it also depends on the way in which the points are distributed. We determine conditions on the function f to ensure the convergence of the polynomials pn(x) to the function f, as the continuity of the function is not enough. The question for analytic functions is answered using potential theory. Convergence and divergence rate of interpolants of analytic functions on the interval are investigated. We also study a generalized Runge phenomenon and find out how the location of the points and poles affect the convergence.