Interpolation sequences and nonspanning systems of exponentials on curves

Gaisin, Rashit Akhtyarovich

doi:10.1070/sm9370

Cited by 3 publications

(1 citation statement)

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“…In a series of papers, there was found a close relation between the regularity of the growth of the sum of the Dirichlet series (1.2) on 𝛾 ∈ Γ with the incompletness of the system of exponentials {︀ 𝑒 𝜆𝑛𝑧 }︀ on the arcs 𝛾 ′ ⊂ 𝛾 and especially with a strong incompletness of this exponential system in a vertical strip, see [7]- [9]. It should be noted that the results of works [8], [9] on the incompletness of the system {︀ 𝑒 𝜆𝑛𝑧 }︀ on the arcs can be applied to studying the uniqueness theorems and asymptotic properties of entire Dirichlet series (1.2) with no restrictions for the growth 𝑀 𝐹 (𝜎), that is, in the most general case.…”

Section: Introductionmentioning

confidence: 98%

Behavior of entire Dirichlet series of class $\underline{D}(\Phi)$ on curves of bounded $K$-slope

Aitkuzhina,

Gaisin,

Gaisin

2023

Ufimsk. Mat. Zh.

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We study an asymptotic behavior of the sum of an entire Dirichlet series 𝐹 (𝑠) = ∑︀ 𝑛 𝑎 𝑛 𝑒 𝜆𝑛𝑠 , 0 < 𝜆 𝑛 ↑ ∞, on curves of a bounded 𝐾-slope naturally going to infinity. For entire transcendental functions of finite order having the formPólya showed that if the density of the sequence {𝑝 𝑛 } is zero, then for each curve 𝛾 going to infinity there exists an unbounded sequence {𝜉 𝑛 } ⊂ 𝛾 such that, as 𝜉 𝑛 → ∞, the relation holds:is the maximum of the absolute value of the function 𝑓 . Later these results were completely extended by I.D. Latypov to entire Dirichlet series of finite order and finite lower order according in the Ritt sense. A further generalization was obtained in works by N.N. Yusupova-Aitkuzhina to more general classes 𝐷(Φ) and 𝐷(Φ) defined by the convex majorant Φ. In this paper we obtain necessary and sufficient conditions for the exponents 𝜆 𝑛 ensuring that the logarithm of the absolute value of the sum of any Dirichlet series from the class 𝐷(Φ) on the curve 𝛾 of a bounded 𝐾-slope is equivalent to the logarithm of the maximum term as 𝜎 = Re 𝑠 → +∞ over some asymptotic set, the upper density of which is one. We note that for entire Dirichlet series of an arbitrarily fast growth the corresponding result for the case of 𝛾 = R + was obtained by A.M. Gaisin in 1998.

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