Fr\'echet spaces of general Dirichlet series

TL;DR

This paper investigates the topological and geometrical properties of Fréchet spaces of general Dirichlet series, extending previous work on ordinary Dirichlet series to more general frequency sequences and convergence conditions.

Contribution

It introduces an abstract framework for Fréchet spaces of λ-Dirichlet series, encompassing various convergence and boundedness conditions, and connects these spaces to Fourier coefficients on λ-Dirichlet groups.

Findings

Developed an abstract setting for pre-Fréchet spaces of λ-Dirichlet series.

Defined new classes of Fréchet spaces based on admissible norms and convergence abscissas.

Connected Dirichlet series spaces to Fourier analysis on λ-Dirichlet groups.

Abstract

Inspired by a recent article on Fr\'echet spaces of ordinary Dirichlet series $\sum a_n n^{-s}$ due to J.~Bonet, we study topological and geometrical properties of certain scales of Fr\'echet spaces of general Dirichlet spaces $\sum a_n e^{-\lambda_n s}$. More precisely, fixing a frequency $\lambda = (\lambda_n)$, we focus on the Fr\'echet space of $\lambda$-Dirichlet series which have limit functions bounded on all half planes strictly smaller than the right half plane $[\mathrm{Re} >0]$. We develop an abstract setting of pre-Fr\'echet spaces of $\lambda$-Dirichlet series generated by certain admissible normed spaces of $\lambda$-Dirichlet series and the abscissas of convergence they generate, which allows also to define Fr\'echet spaces of $\lambda$-Dirichlet series for which $a_n e^{-\lambda_n/k}$ for each $k$ equals the Fourier coefficients of a function on an appropriate $\lambda$-Dirichlet group.