Chapter 6: Rational Approximation Theory

By definition, a rational function is the quotient of two polynomials; and by Theorem 4.3.1, in equivalent formulation, it is a function meromorphic on all of . The poles and zeros may not accumulate in , and thus there are finitely many as a consequence of Bolzano–Weierstrass (Theorem 1.1.2).

When we refer to approximation, we refer to the approximation of a function as the uniform limit (of a sequence) of functions. Let be compact and suppose is a given function on . As a consequence of Mergelyan’s Theorem (Theorem 6.2.2), sufficient conditions for to be the uniform limit of rational functions whose poles lie in (a subset of given points of) are the continuity of on and the holomorphy of on .