Polynomial and Rational Approximation—Runge Theory

Abstract

The power of “integral representations” was demonstrated in 3.8: they lead to local power series expansions. If the (Riemann) integrals involved are approximated by Riemann sums, we get global rational approximations to the function. (See 8.8 below.) If further the domains are properly disposed in ℂ, the “poles” in these rational functions can be “shoved to infinity” and the rational functions thereby approximated by polynomials. Global polynomial approximants are thus produced. They are a very powerful tool for investigating holomorphic functions: often a theorem is easy for polynomials and remains valid under local uniform convergence, hence its validity passes over to holomorphic unctions generally.

An erratum to this chapter is available at http://dx.doi.org/10.1007/978-3-0348-9374-9_18