LAUSR

Given a frequency λ = (λn) and l ≥ 0, we introduce the scale of Banach spaces H ∞,l[Re > 0] of holomorphic functions f on the right half-plane [Re > 0], which satisfy (A) the growth condition |f(s)| = O((1+ |s|)l), and (B) are such that they on some open subset and for some m ≥ 0 coincide with the pointwise limit (as x → ∞) of the so-called (λ,m)-Riesz means ∑ λn 0 of some λ-Dirichlet series ∑ ane n. Reformulated in our terminology, an important result of M. Riesz shows that in this case the function f for every k > l is the pointwise limit of the (λ, k)-Riesz means of D. Our main contribution is an extension – showing that ’after translation’ every bounded set in H ∞,l[Re > 0] is uniformly approximable by all its (λ, k)-Riesz means of order k > l. This follows from an appropriate maximal theorem, which in fact turns out to be at the very heart of a seemingly interesting structure theory of the Banach spaces H ∞,l[Re > 0]. One of the many consequences is that H ∞,l[Re > 0] basically consists of those holomorphic functions on [Re > 0], which fulfill the growth condition |f(s)| = O((1+ |s|)l) and are of finite uniform order on all abscissas [Re = σ], σ > 0. To establish all this and more, we need to reorganize (and to improve) various aspects and keystones of the classical theory of Riesz summability of general Dirichlet series as invented by Hardy and M. Riesz.