Holomorphic functions of finite order generated by Dirichlet series

TL;DR

This paper introduces a new framework for analyzing holomorphic functions generated by Dirichlet series, establishing approximation properties and a structure theory for associated Banach spaces based on Riesz means.

Contribution

It extends classical Riesz summability theory by showing uniform approximation of functions in specific Banach spaces using Riesz means after translation.

Findings

Bounded sets in the Banach spaces are uniformly approximable by Riesz means.

The spaces consist of functions with a Riesz germ and finite uniform order.

A maximal theorem underpins the structure theory of these Banach spaces.

Abstract

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.