Adam Carlén: Herglotz–Nevanlinna functions and rational approximations

Abstract.

Herglotz–Nevanlinna functions maps the complex upper half plane analytically to itself. The classic theory by Nevanlinna provides an integral representation for such functions in terms of a positive Borel measure on the real line, and establish a one-to-one correspondence between the func- tions and the class of finite positive Borel measures, D(R). The first part of this thesis utilize some techniques from functional analysis to prove this in detail. The second part of this thesis is devoted to show an approx- imation theorem for Herglotz–Nevanlinna functions in terms of rational functions with poles of order at most one. Any such function is again a Herglotz–Nevanlinna function. It turns out that the extreme points of D(R) are precisely the point mass measures δx0 , and by establishing com- pactness and convexity of D(R) in the weak* sense, the Krein–Milman theorem may be applied to extract a pointwise convergent sequence of such rational functions