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On some extremal problems for entire functions of exponential type

https://doi.org/10.22405/2226-8383-2025-26-1-47-61

Abstract

In this paper we consider a number of extremal problems for nonnegative and integrable entire functions of exponential type ⩽ 𝜎 (the class ℰ+1,𝜎). The problems under consideration have the following form. Let Λ𝜌 be a translation invariant operator with a locally integrable symbol 𝜌(𝑥), 𝑥 ∈ R, such that 𝜌(𝑥) = 𝜌(−𝑥), 𝑥 ∈ R. For a fixed 𝜎 > 0, it is required to find the following constants:

This general problem reduces to an equivalent extremal problem for positive-definite functions, the solution of which is known. As consequence, we obtained exact values of 𝑀*(𝜌, 𝜎) and 𝑚*(𝜌, 𝜎) for a number of different symbols 𝜌. In particular, we consider cases where Λ𝜌 is a differential or difference operator of a special form.

About the Author

Anatoliy Dmitrievich Manov
Donetsk State University (Donetsk); Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation

candidate of physical and mathematical sciences



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Review

For citations:


Manov A.D. On some extremal problems for entire functions of exponential type. Chebyshevskii Sbornik. 2025;26(1):47-61. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-1-47-61

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