Weight Index of Concentration

Gaisin, A. M. , Gaisin, R. A.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 1.

Abstract:
In terms of the concentration weight index, the behavior of the function \(|W(re^{i\theta})|^{-1}\) is investigated as \(\theta \to 0\), where \(W\) is an even entire function of exponential type that has only real zeros. This question is relevant in a number of problems of complex analysis related to the strongly nonspanning (strongly minimality) of a system of exponentials on a family of curves, Pavlov-Korevar-Dixon interpolation, and analytic continuation of limit functions of sequences of polynomials from exponentials. This circle of problems goes back to the following problem of A. F. Leontief posed in 1956: under what conditions \(\sup\nolimits_{\theta \ne 0, \pi} H(\theta) < \infty,\) where \(H(\theta)\) is the indicatrix (indicator) of the function \(W^{-1}(\lambda)\), \(\lambda = re^{i\theta}\). In the works of A. F. Leontief and A. Baillette, some estimates for this indicator were obtained, but they turned out to be very rough. For arbitrary entire functions of proximate order, I. F. Krasichkov in 1965 proved a theorem that answers A. F. Leontief's question. As was shown, a necessary and sufficient condition for the finiteness of the indicator  \(H(\theta)\) is the finiteness of the concentration index of the sequence \(\Lambda\) of zeros of the entire function \(W\), calculated through the growth function for a given proximate order. Of particular interest is the case when the sequence  \(\Lambda\) is an interpolation sequence. In this case, as shown by B. Berndtsson, the comparison function is some concave majorant from the convergence class. However, this function (i.e., the weight) does not have to have the regular variation at infinity. Therefore, this case is considered in the present paper. The main result: in order for the weight lower indicator \(H(\omega, \theta)\) of the function \(W\) to be uniformly bounded below, it is necessary and sufficient that the concentration weight index \(I_\Lambda(\omega, \mathbb{R})\) be finite.

Keywords: entire function, lower indicator, coindicatrix, maximum density, weight index of concentration

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