Arne Beurling

Conformal invariants and function-theoretic null-sets

The most useful conformal invar iants are obtained by solving conformMly invar ian t ex t remal problems. The i r usefulness derives f rom the fac t tha t they must automat ical ly satisfy a principle of majorizat ion. There is a r ich variety of such problems, and if we would aim at completeness this paper would assume forbidding proport ions. We shall therefore l imit ourselves to a few part icular ly simple i n v a r i a n t s and study thei r propert ies and in ter re la t ions in considerable detail. Each class of invar ian ts is connected wi th a category of null-sets, which by this very fac t en ter na tura l ly in funct iontheoret ic considerat ions. A null-set is the complement of a region for which a cer ta in conformal invar iant degenerates. Inequal i t ies between invar iants lead to inclusion relat ions between the corresponding classes of null-sets. Th roughou t this paper Y2 will denote an open region in the extended z plane, and Zo will be a dis t inguished point in t~. Most results will be formula ted for the case z 0 ~ c~, but the t rans i t ion to z o = c~ is always trivial. In some instances the la t te r case offers formal advantages. We shall consider classes of funct ions f(z) which are analyt ic and singlevalued in some region t). F o r a general class ~ the region t~ is al lowed to vary with f , but the subclass of funct ions in a fixed region t~ will be denoted by ~(t2). For ZoE ~ we in t roduce the quant i ty

On the closure of characters and the zeros of entire functions

The problem to be studied in this paper concerns the closure properties on an interval of a set of characters {e~nx}~, where A = {2n}~ is a given set of real or complex numbers without finite point of accumulation. This problem is for obvious reasons depending on the distribution of zeros of certain entire functions of exponential type. The main problem of the paper is to determine the closure radius Q = Q(A)defined as the upper bound of numbers r such that (ei~x)~EA span the space L 2 ( r , r ) . The value of r does not change if a finite number of points are removed from or adjoined to A. Nor does Q(A) change if the metric in the previous definition is replaced by any other LV-metric, or by a variety of other topologies. I f A contains complex numbers we shall always assume (1)< 6~t ~ (0.1) 9 ~eA ~