Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 3220–3258 | Cite as

\(\pmb {{\mathcal {H}}_{p}}\)-Theory of General Dirichlet Series

  • Andreas Defant
  • Ingo SchoolmannEmail author


Inspired by results of Bayart on ordinary Dirichlet series \(\sum a_n n^{-s}\), the main purpose of this article is to start an \({\mathcal {H}}_p\)-theory of general Dirichlet series \(\sum a_n e^{-\lambda _{n}s}\). Whereas the \({\mathcal {H}}_p\)-theory of ordinary Dirichlet series, in view of an ingenious identification of Bohr, may be seen as a sub-theory of Fourier analysis on the infinite dimensional torus \({\mathbb {T}}^\infty \), the \({\mathcal {H}}_p\)-theory of general Dirichlet series is build as a sub-theory of Fourier analysis on certain compact abelian groups, including the Bohr compactification \({\overline{{\mathbb {R}}}}\) of the reals. Our approach allows to extend various important facts on Hardy spaces of ordinary Dirichlet series to the much wider setting of \({\mathcal {H}}_p\)-spaces of general Dirichlet series.


General Dirichlet series Hardy spaces Bohr compactification 

Mathematics Subject Classification

Primary 43A17 Secondary 30H10 30B50 



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikCarl von Ossietzky UniversitätOldenburgGermany

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