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24 June 2024
 
  » arxiv » 2302.00231

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Projection constants for spaces of Dirichlet polynomials
Andreas Defant ; Daniel Galicer ; Martín Mansilla ; Mieczysław Mastyło ; Santiago Muro ;
Date 1 Feb 2023
AbstractGiven 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 $oldsymbol{lambda}ig(mathcal{H}_infty^{J}(omega) ig)$ of the finite dimensional Banach space $mathcal{H}_infty^{J}(omega)$ equipped with the norm $|D|= sup_{ ext{Re},s>0} |D(s)|$. Based on harmonic analysis on $omega$-Dirichlet groups, we prove the formula $ oldsymbol{lambda}ig(mathcal{H}_infty^{J}(omega) ig) = lim_{T o infty} frac{1}{2T} int_{-T}^T Big|sum_{n in J} e^{-iomega_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 $mathcal{H}_infty^{leq x}ig( (log n)ig)$ of all ordinary Dirichlet polynomials $D(s) = sum_{n leq x} a_n n^{-s}$ of length $x$ show the asymptotically correct order $ oldsymbol{lambda}ig(mathcal{H}_infty^{leq x}ig( (log n)ig)ig) sim sqrt{x}/(log log x)^{frac{1}{4}}. $
Source arXiv, 2302.00231
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