On an extremal problem for positive definite functions
https://doi.org/10.22405/2226-8383-2021-22-5-161-171
Abstract
In this paper we consider an extremal problem related to a set of continuous positive definite functions on R whose support is contained in the closed interval [−𝜎, 𝜎], 𝜎 > 0 and the value
at the origin is fixed (the class F𝜎).
We consider the following problem. Let 𝜇 be a linear locally bounded functional on the set of continuous functions which have compact support, i.e. 𝐶𝑐(R) and suppose that 𝜇 is real-valued functional on the sets F𝜎, 𝜎 > 0. For a fixed 𝜎 > 0, it is required to find the following constants:
𝑀(𝜇, 𝜎) := sup {𝜇(𝜙) : 𝜙 ∈ F𝜎} , 𝑚(𝜇, 𝜎) := inf {𝜇(𝜙) : 𝜙 ∈ F𝜎} .
We have obtained a general solution to this problem for functionals of the following form 𝜇(𝜙) = ∫︀ R 𝜙(𝑥)𝜌(𝑥)𝑑𝑥, 𝜙 ∈ 𝐶𝑐(R), where 𝜌 ∈ 𝐿𝑙𝑜𝑐(R) and 𝜌(𝑥) = 𝜌(−𝑥) a.e. on 𝑥 ∈ R.
For 𝜌(𝑥) ≡ 1, the value of 𝑀(𝜇, 𝜎) was obtained by Siegel in 1935 and, independently, by Boas and Kac in 1945. In this article, we have obtained explicit solution to the problem under consideration in cases of 𝜌(𝑥) = 𝑖𝑥, 𝜌(𝑥) = 𝑥2 and 𝜌(𝑥) = 𝑖 sign 𝑥, 𝑥 ∈ R.
In addition, in this paper we study the connection between the problem under consideration and pointwise inequalities for entire functions of exponential type 6 𝜎 whose restrictions on R
are in 𝐿1(R). In particular, sharp inequalities are obtained for the first and second derivatives of such functions.
About the Author
Anatoliy Dmitrievich Manov
Donetsk National University
Ukraine
Ukraine
candidate of physical and mathematical sciences
References
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Review
For citations:
Manov A.D. On an extremal problem for positive definite functions. Chebyshevskii Sbornik. 2021;22(5):161-171. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-5-161-171
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