Introduction

If Ω is an open set in C or Rn(n ≥ 2), then we will use Ω* to denote the Alexandroff, or one-point, compactification of Ω, and will use A to denote the ideal point. Thus Ω* = Ω ∪ {A}, and a set A is open in Ω* if either A is an open subset of Ω or A = Ω*\K, where K is a compact subset of Ω. In the special case where Ω is C or Rn we continue to write ∞ for A.

If A is a subset of C, we denote by Hol(A) the collection of all functions which are holomorphic on an open set containing A. Historically the following result (essentially in [Run]; cf. [Con, pp.198, 201]) can be regarded as the starting point of the theory of holomorphic approximation.

Runge's Theorem (1885).Let Ω be an open subset ofCand K be a compact subset of Ω. The following are equivalent:

1. (a) for each f in Hol(K) and each positive number ∈, there exists g in Hol(Ω) such that |g – f| < ∈ on K;

2. (b) Ω*\K is connected.

Condition (b) above is equivalent to asserting that no component of Ω\K is relatively compact in Ω. Also, when Ω = C, this condition is clearly equivalent to saying that C\K is connected.

We record below one further important development in the theory of holomorphic approximation, which deals with approximation of a much larger class of functions on a given compact set K.

Introduction

Many of the results in the preceding chapters have superharmonic analogues, some of which we will consider in this chapter. Thus, for example, we will examine which pairs (Ω, E) have the property that every u in S(E) can be uniformly approximated on E by functions in S(Ω). However, in the case of superharmonic functions, it may be possible not only to approximate, but even to extend, functions in S(E).

For example, suppose that K is a compact subset of an open set Ω such that Ω*\K is connected. Then, as we saw in Theorem 1.7, for every u in H(K) and every positive number ∈, there exists υ in H(Ω) such that |υ – u| < ∈ on K. The corresponding fact for superharmonic functions (a special case of Theorem 6.1 below) is that, for every u in S(K) there exists v in S(Ω) such that v = u on K.

Strong Extension

We begin the chapter with some such extension results. Later we will deal with Runge and Arakelyan approximation. Throughout this chapter Ω denotes an open set in Rn and E is a relatively closed subset of Ω. By a continuous superharmonic function we mean one which is both finite-valued and continuous.

We call (Ω, E) an extension pair for superharmonic functions (resp. for continuous superharmonic functions) if, for each function (resp. each continuous function) u in S(E) there exists υ in S(Ω) (resp. in C(Ω)∩S(Ω)) such that υ = u on E.

Introduction

This book differs essentially from the existing monographs on countable Markov chains. It intends to be, on the one hand, much more constructive than books similar to, for example Chung's [Chu67] and, on the other hand, much less constructive than some elementary monographs on queueing theory, where the emphasis is mainly put on the derivation of explicit expressions. The method of generating functions, which is to be sure the most constructive approach, is not included, since the dimension of the problems it can solve is small (in general ≤ 2). Our book could equally be called Constructive use of Lyapounov functions method. Here the term constructive is taken in the sense close to the one widely accepted in constructive mathematical physics. One can say that the objects considered have a sufficiently rich structure to be concrete, although the results may not always be explicit enough, as commonly understood. Semantically, it is permissible to say that our methods are more qualitative constructive than quantitative constructive.

The main goal of the book is to provide methods allowing a complete classification (necessary and sufficient conditions) or, in other words, allowing us to say when a Markov chain is ergodic, null recurrent or transient. Moreover, it turns out that, without doing much additional work, it is possible to study the stability (continuity or even analyticity) with respect to parameters, the rate of convergence to equilibrium, …, etc. by using the same Lyapounov functions.

Analytic capacity and removable sets

In this chapter we shall discuss a classical problem in complex analysis and its relations to the rectifiability of sets in the complex plane C. The problem is the following: which compact sets E ⊃ C are removable for bounded analytic functions in the following sense?

(19.1) If U is an open set in C containing E and f: U\E → C is a bounded analytic function, then f has an analytic extension to U.

This problem has been studied for almost a century, but a geometric characterization of such removable sets is still lacking. We shall prove some partial results and discuss some other results and conjectures. For many different function classes a complete solution has been given in terms of Hausdorff measures or capacities. For example, if the boundedness is replaced by the Holder continuity with exponent α, 0 < α < 1, then the necessary and sufficient condition for the removability of E is that H1+α(E) = 0, see Exercise 4, Dolzenko [1] and Uy [2], and for the corresponding question for harmonic functions Carleson [1]. Král [1] proved that for the analytic BMO functions the removable sets E are characterized by the condition H1(E) = 0. The problem (19.1) is more delicate, because the metric size is not the only thing that matters; the rectifiability structure also seems to be essential as we shall see.

Ahlfors [1] introduced a set function γ, called analytic capacity, whose null-sets are exactly the removable sets of (19.1).