Projection constants for spaces of Dirichlet polynomials

TL;DR

This paper derives asymptotically precise estimates for the projection constants of spaces of Dirichlet polynomials using harmonic analysis, with applications to classical number theory problems.

Contribution

It provides a new formula for the projection constant of Dirichlet polynomial spaces and applies harmonic analysis to obtain asymptotic estimates, including for classical Dirichlet polynomials.

Findings

Derived a formula for the projection constant involving an integral limit.

Applied the formula to specific frequency sequences and sets.

Obtained asymptotic order for Dirichlet polynomial spaces related to number theory.

Abstract

Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in \mathbb{C}$. The main aim is to prove asymptotically correct estimates for the projection constant $\boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big)$ of the finite dimensional Banach space $\mathcal{H}_\infty^{J}(\omega)$ equipped with the norm $\|D\|= \sup_{\text{Re}\,s>0} |D(s)|$. Based on harmonic analysis on $\omega$-Dirichlet groups, we prove the formula $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{J}(\omega) \big) = \lim_{T \to \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 $J$. 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 $ \boldsymbol{\lambda}\big(\mathcal{H}_\infty^{\leq x}\big( (\log n)\big)\big) \sim \sqrt{x}/(\log \log x)^{\frac{1}{4}}. $