Projection constants for spaces of Dirichlet polynomials

Abstract

Given a frequency sequence $\omega =(\omega _n)$ and a finite subset $J \subset {\mathbb {N}}$, we study the space ${\mathscr {H}}_{\infty }^{J}(\omega )$ of all Dirichlet polynomials $D(s):= \sum \nolimits _{n \in J} a_n e^{-\omega _n s}, \, s \in {\mathbb {C}}$. The main aim is to prove asymptotically correct estimates for the projection constant $\varvec{\lambda }\big ({\mathscr {H}}_\infty ^{J}(\omega ) \big )$ of the finite dimensional Banach space ${\mathscr {H}}_\infty ^{J}(\omega )$ equipped with the norm $\Vert D\Vert = \sup _{\text {Re}\,s>0} |D(s)|$. Based on harmonic analysis on $\omega$-Dirichlet groups, we prove the formula $\varvec{\lambda }\big ({\mathscr {H}}_\infty ^{J}(\omega ) \big ) ~ = ~ \displaystyle \lim _{T \rightarrow \infty } \frac{1}{2T} \int _{-T}^T \Big |\sum _{n \in J} e^{-i\omega _n t}\Big |\,dt,$ and apply it to various concrete frequencies $\omega$ and index sets J. Combining with a recent deep result of Harper from probabilistic analytic number theory, we for the space ${\mathscr {H}}_\infty ^{\le x}\big ( (\log n)\big )$ of all ordinary Dirichlet polynomials $D(s) = \sum _{n \le x} a_n n^{-s}$ of length x show the asymptotically correct order $\varvec{\lambda }\big ({\mathscr {H}}_\infty ^{\le x}\big ( (\log n)\big )\big ) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}.$