Runge's Phenomenon - Introduction

Introduction

The Weierstrass approximation theorem states that every continuous function f(x) defined on an interval can be uniformly approximated as closely as desired by a polynomial function Pn(x) of degree ≤ n, i.e.,

Interpolation at equidistant points is a natural and well-known approach to construct approximating polynomials. Runge's phenomenon demonstrates, however, that interpolation can easily result in divergent approximations.

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