The most important An-modules are the holonomic modules, also known among PDE theorists as maximally overdetermined systems. An An-module is holonomic if it has dimension n. Ordinary differential equations with polynomial coefficients correspond to holonomic modules. In this chapter we begin the study of holonomic modules, which will be one of the central topics of the second half of the book.

DEFINITION AND EXAMPLES.

A finitely generated left An-module is holonomic if it is zero, or if it has dimension n. Recall that by Bernstein's inequality this is the minimal possible dimension for a non-zero An-module. We already know an example of a holonomic An- module, viz. K[X] = K[x1, …, xn]. We also know that An itself is not a holonomic module: it has dimension 2n.

It is easy to construct holonomic modules if n = 1. Let I ≠ 0 be a left ideal of A1. By Corollary 9.3.5, d(A1/I) ≤ 1. If I ≠ A1 then, by Bernstein's inequality, d(A1/I) = 1. Hence A1/I is a holonomic A1-module.

This is wonderful source of examples, which will pour forth with the help of the next proposition.

Proposition. Let n be a positive integer.

1. Submodules and quotients of holonomic An-modules are holonomic.

2. Finite sums of holonomic An-modules are holonomic.

Proof: (1) These follow from Bernstein's inequality. Let M be a left An module, and N a submodule of M. From Theorem 9.3.2, d(N) ≤ d(M) and d(M/N) ≤ d(M). Since d(M) = n, and using Bernstein's inequality, we deduce that d(N) = d(M/N) are also equal to n. Thus N and M/N are holonomic. Now (2) follows from Corollary 9.3.3 and (1).

In sections 1.1, 1.2 and 1.3 of this chapter, we briefly introduce basic notions and some results borrowed from the theory of discrete time homogeneous countable Markov chains (MC).

In section 1.4, some well known examples of MCs are given, for which a complete classification can be obtained by elementary methods: simple probabilistic arguments in 1.4.1, explicit solution of recurrent equations in 1.4.2, generating functions in 1.4.3.

It is not our intention to devote a detailed section to the fundamentals of probability theory, which are presented in a plethora of excellent text-books. Thus, we only introduce in fact the minimal basic notions and notation useful for our purpose.

• The events are the subsets of some abstract set Ω, which belong to Σ, the σ-algebra defined on Ω.

• The couple (Ω, Σ) is a measurable space and the sets belonging to Σ are

Σ-measurable sets.

• The triple (Ω, Σ, µ), where µ, is a positive measure defined on Σ, is a measure space. A probability space is a measure space of total measure 1, i.e. µ(Σ) = 1, and in this case most of the time we shall write (Ω, Σ, P).

• A Σ-measurable real-valued function f with domain Ω is called a random variable. More generally a random element ϕ with values in a measurable space (X, B) is a measurable mapping of (Ω, Σ, P) into (X, B). For X = RN or ZN, B being the σ-algebra of Borel sets, we shall speak of random vectors.

The year 1885 has a special significance in the history of approximation theory. It was then that Weierstrass published his famous result which says that a continuous function on a closed bounded interval of the real line can be uniformly approximated by polynomials. The same year saw the birth of holomorphic approximation in the celebrated paper of Runge [Run]. Given an open set Ω in the complex plane C, which compact subsets K have the property that any holomorphic function defined on a neighbourhood of K can be uniformly approximated on K by functions holomorphic on Ω? Runge's Theorem supplies the answer: precisely the sets K such that Ω\K has no components which are relatively compact in Ω. Since Runge's original work holomorphic approximation has developed into a significant research area. We mention particularly the contributions of Carleman [CarT], Alice Roth [Rot1], [Rot3], Mergelyan [Mer], Arakelyan [Ara1] and Nersesyan [Ner]. A helpful account of these and other results can be found in the book by Gaier [Gai]. The purpose of these notes is to give a corresponding account of the theory of harmonic approximation in Euclidean space Rn (n ≥ 2).

The starting point in the history of harmonic approximation is not as easy to identify. In the case of approximation in higher dimensions, the paper of Walsh [Wal] in 1929 seems a reasonable choice, but for approximation in the plane mention must also be made of work of Lebesgue [Leb] in 1907.

Introduction

The following is Arakelyan's generalization of Mergelyan's Theorem (see §1.1) to non-compact sets. It can be found in [Ara1] or [Ara2].

Arakelyan's Theorem (1968).Let Ω be an open set in C and E be a relatively closed subset of Ω. The following are equivalent:

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

2. (b) Ω*\E is connected and locally connected.

The above local connectedness condition will be discussed in §3.2. Its first appearance (at least, in an equivalent form) in the context of holomorphic approximation occurs in early work of Alice Roth which is not as well known as it should be. It is remarkable that, as early as 1938, Roth [Rot1] had shown that (when Ω = C) condition (b) above is sufficient for uniform approximation of functions in Hol(E) by entire holomorphic functions. (See [Rot2] for the generalization to other choices of Ω.) Of course, Arakelyan's Theorem is an improvement of Roth's result.

This chapter presents corresponding results for uniform approximation by harmonic functions on relatively closed sets. In fact, we will obtain generalizations of Theorems 1.3, 1.7, 1.10, 1.15, and Corollary 1.16. Further, it will be shown that, whenever uniform approximation is possible, something rather better is also true (at least, in most cases). The main results are Theorems 3.15, 3.17 and 3.19.