Projection constants for spaces of Dirichlet polynomials

2023/02/01 by Andreas Defant, Defant, Andreas, Daniel Galicer +7
Computer Science · Mathematics · #43A77 #46B06. Secondary: 46B07 #46B07 #46G25 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Number Theory (math.NT) #Primary: 30B50

paper · pdf · doi:10.48550/arxiv.2302.00231

openalex publication_date 2023/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a frequency sequence ω=(ω_n) and a finite subset J ⊂ ℕ, we study the space H_∞^J(ω) of all Dirichlet polynomials D(s) := ∑_{n ∈ J} a_n e^{-ω_n s}, s ∈ ℂ. The main aim is to prove asymptotically correct estimates for the projection constant \boldsymbolλ(H_∞^J(ω) ) of the finite dimensional Banach space H_∞^J(ω) equipped with the norm ‖D‖= sup_{Re s>0} |D(s)|. Based on harmonic analysis on ω-Dirichlet groups, we prove the formula \boldsymbolλ(H_∞^J(ω) ) = lim_{T → ∞} (1)/(2T) ∫_-T^T |∑_{n ∈ J} e^{-iω_n t}| 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 H_∞^{≤ x}( (log n)) of all ordinary Dirichlet polynomials D(s) = ∑_{n ≤ x} a_n n^-s of length x show the asymptotically correct order \boldsymbolλ(H_∞^{≤ x}( (log n))) ∼ √(x)/(log log x)^(1)/(4).

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• Projection constants for spaces of multivariate polynomials
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• Holomorphic functions of finite order generated by Dirichlet series
• Maximal ideal space of some Banach algebras of Dirichlet series
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