Projection constants for spaces of Dirichlet polynomials

TL;DR

Based on harmonic analysis on ω -Dirichlet groups, the formula λ ¡ H J ∞ ( ω ) equipped with the norm k D k = sup Re s > 0 | D ( s ) is proved.

Abstract

Given a frequency sequence $ω=(ω_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(ω)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-ω_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(ω)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $ω$-Dirichlet groups, we prove the formula $ \boldsymbolλ\big(\mathcal{H}_\infty^{J}(ω) \big) = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^T \Big|\sum_{n \in J} e^{-iω_n t}\Big|\,dt\,, $ and apply it to various concrete frequencies $ω$ 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 $\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)$ of all ordinary Dirichlet polynomials $D(s) = \sum_{n \leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ \boldsymbolλ\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $

Theorems & Definitions (40)

• Theorem 2.1
• Theorem 2.2
• proof : Proof of Theorem \ref{['rudy']}
• proof : Proof of Theorem \ref{['C(G)proj']}
• Corollary 2.3
• proof
• Remark 2.4
• Corollary 2.5
• Proposition 2.6
• proof