David H. Armitage

A NEW TYPE OF CESÀRO MEAN

For a given sequence λ = (λη ) with 0 < λ„ < λ „ + 1 —> oo we introduce a new summability method Cx which provides a generalization of the classical (C, 1) method. Abelian and Tauberian theorems for Cx are established, the main Tauberian result being of slowly decreasing character. AMS-Classification: 40E05

Universal overconvergence of polynomial expansions of harmonic functions

For each compact subset K of RN let H(K) denote the space of functions that are harmonic on some neighbourhood of K. The space H(K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of RN such that 0 ∈ Ω and RN\ ??Ω is connected. It is shown that there exists a series ∑ Hn, where Hn is a homogeneous harmonic polynomial of degree n on RN such that (i) ∑Hn converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑Hn are dense in H(K) for every compact subset K of RN\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ℝm×ℝn, wherem⩾2 andn⩾2, such thatu(x, .) is subharmonic on ℝn for each fixedx in ℝm andu(.,y) is subharmonic on ℝm for each fixedy in ℝn. We give a local integrability condition which ensures the subharmonicity ofu on ℝm×ℝn, and we show that this condition is close to being sharp. In particular, the local integrability of (log+u+)m+n−2+α is enough to secure the subharmonicity ofu if α>0, but not if α<0.

Discrete Abel means

Corresponding to each sequence λ = (λη) with 1 < λ„ < λ„+ ι —• oo we introduce a new summability method Ax (the discrete Abel mean) which is related to, but stronger than, the classical method of Abel. We prove Abelian and Tauberiem theorems for Αχ ; our main Tauberian condition concerns slow decrease. AMS-Classification: 40E05

Uniform and tangential harmonic approximation

We give a survey of Runge-type harmonic approximation theorems and the techniques used to prove them. The emphasis is on generalizations of Walsh’s classical theorem concerning uniform harmonic approximation on compact sets to the case where approximation takes place on certain non-compact sets: both uniform and tangential approximation are treated. We also give some applications of the theory to the construction of harmonic functions exhibiting various kinds of unexpected behaviour. The course is partly intended to provide preparatory material for S. J. Gardiner’ course “Harmonic approximation and applications”, published in this volume.

CHARACTERIZATIONS OF THIN AND SEMI-THIN SETS

Under mild assumptions about a function φ : (0,1] -+ (0,+°°), sets E which are thin at the origin 0 of R m (m > 3) are characterized by the existence of a Newtonian potential u for which lim inf <(>(|x|)u(x) > 0 and x-K),xeE t if>(t)M(u,t)dt < where M(u,t) is the mean value of u on the sphere 0 of centre 0 and radius t. Various generalizations and extensions are indicated. AMS-Classification: 31B15

The Martin Boundary

We saw in Chapter 1 that if μ is a measure on S, then the equation

Polar Sets and Capacity

h\left( x \right) = \int_S {K\left( {x,y} \right)d\mu \left( y \right)} \left( {x \in B} \right),