Projection constants for spaces of Dirichlet polynomials

Abstract

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

arXiv admin note: substantial text overlap with arXiv:2208.06467