Projection constants for spaces of Dirichlet polynomials

Abstract

Given a frequency sequence ω = (ωn) and a finite subset J ⊂ N, we study the space H J ∞(ω) of all Dirichlet polynomials D(s) := Σ n∈J ane −ωns , s ∈ C. The main aim is to prove asymptotically correct estimates for the projection constant λ ¡ H J ∞(ω) ¢ of the finite dimensional Banach space H J ∞(ω) equipped with the norm kDk = supRes>0 |D(s)|. Based on harmonic analysis on ω-Dirichlet groups, we prove the formula λ ¡ H J ∞(ω) ¢ = lim T→∞ 1 2T ZT −T │Σn∈J e −iωn t ¯ ¯ ¯d t , 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 ∞ ¡ (logn) ¢ of all ordinary Dirichlet polynomials D(s) = P n≤x ann −s of length x show the asymptotically correct order λ ¡ H ≤x ∞ ¡ (logn) ¢¢ ∼ p x/(loglog x) 1 4 .