Holomorphic functions of finite order generated by Dirichlet series

Given a frequency \(\lambda = (\lambda_n)\) and \(\ell \ge 0\), we introduce the scale of Banach spaces \(H_{\infty,\ell}^{\lambda}[Re > 0]\) of holomorphic functions \(f\) on the open right half-plane \([Re > 0]\), which satisfy \((A)\) the growth condition \(|f(s)| = O((1 + |s|)^\ell)\), and \((B)\) have a Riesz germ, i.e. on some open subset and for some \(m \ge 0\) the function \(f\) coincides with the pointwise limit (as \(x \to \infty\)) of the so-called \((\lambda,m)\)-Riesz means \(\sum_{\lambda_n < x} a_n e^{-\lambda_n s}\big( 1-\frac{\lambda_n}{x}\big)^m ,\,x >0\) of some \(\lambda\)-Dirichlet series \(\sum a_n e^{-\lambda_n s}\). Reformulated in our terminology, an important result of M. Riesz shows that in this case the function \(f\) for every \(k >\ell\) is the pointwise limit of the \((\lambda,k)\)-Riesz means of \(D\) on \([Re > 0]\). Our main contribution is an extension -- showing that 'after translation' every bounded set in \(H_{\infty,\ell}^{\lambda}[Re > 0]\) is uniformly approximable by all its \((\lambda,k)\)-Riesz means of order \(k>\ell\). 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 \(H_{\infty,\ell}^{\lambda}[Re > 0]\). One of the many consequences is that \(H_{\infty,\ell}^{\lambda}[Re > 0]\) basically consists of those holomorphic functions on \([Re >0]\), which have a Riesz germ and are of finite uniform order \(\ell\) on \([Re >0]\). 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.