Noncomplete systems of exponentials on arcs and Carleman nonquasianalytic classes. II

Гайсин, А. М.; Gaisin, Rashit Akhtyarovich

doi:10.1090/spmj/1375

Cited by 7 publications

(1 citation statement)

References 14 publications

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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: 92%

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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Section: Introductionmentioning

confidence: 92%

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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Extremal problems in nonquasianalytic Carleman classes. Applications

Гайсин¹

2018

Sb. Math.

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Interpolation sequences and nonspanning systems of exponentials on curves

Gaisin¹

2021

Sb. Math.

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Interpolation sequences of the form are investigated, and also the problem of when the system of exponentials is nonspanning on the family of arbitrary rectifiable curves in the uniform norm. In terms of the interpolation nodes (or equivalently, the exponents of the system of exponentials) a criterion for the interpolation problem to be solvable is established and the strong nonspanning property of is proved. This significantly improves some known results, in particular, results due to Korevaar, Dixon and Berndtsson. Bibliography: 23 titles.

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Noncomplete systems of exponentials on arcs and Carleman nonquasianalytic classes. II

Abstract: In terms of the noncompleteness of a system of exponentials, a criterion is established for the nontriviality of the Siddiqi class on an arc of bounded slope all chords of which have slope strictly smaller than one.

Cited by 7 publications

References 14 publications

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

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

Extremal problems in nonquasianalytic Carleman classes. Applications

Interpolation sequences and nonspanning systems of exponentials on curves

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