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DOI: 10.46698/v3523-1431-1350-j Uniqueness Distributions for Entire Functions with Uniform Constraints on Their Growth
Khabibullin, B. N.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 1.
Abstract:
Let \(M=M_{\mathsf{up}}-M_{\mathsf{low}}\) be the difference of subharmonic functions on the complex plane \(\mathbb C\). First, we discuse the following general problem: What are the conditions for the distribution of points \(Z\) on \({\mathbb{C}}\), under which there is an entire nonzero function \(f\) that vanishes on \(Z\) and satisfies the inequality \(|f|\leq e^M\) on \(\mathbb{C}\)? We formulate some known results for the general problem from one of our papers with co-authors. The next step is to discuss a specific problem of when \(M_{\mathsf{up}}=b|\mathrm{Im}|\) is the module of the imaginary part with a numerical multiplier \(b\geq 0\), and \(M_{\mathsf{low}}\) is the Poisson transformation of a positive even function \(w\) on the real axis \({\mathbb{R}}\), increasing on the positive semi-axis \({\mathbb{R}}^+\), and with a finite logarithmic integral. A very significant contribution to this theory is contained in a number of fundamental works by A. V. Abanin, including his known monograph. It is precisely such classes of entire functions that arise after the Fourier--Laplace transform of test functions on compacts. In this direction, the article discusses the limits of applicability of the Beurling--Malliavin theory, and also provides our criterion with co-authors, but only for the zero function \(w=0\). The final main result of the article extends the last criterion to the cases of a nonzero function \(w\neq 0\).
Keywords: entire function, point distribution, zero distribution, subharmonic function, mass distribution, Cartwright class, logarithmic integral, Poisson integral, ultradifferentiable function, ultradistribution
Language: Russian
For citation: Khabibullin, B. N. Uniqueness Distributions for Entire Functions with Uniform Constraints on Their Growth, Vladikavkaz Math. J., 2025, vol. 27, no. 1, pp. 112-126 (in Russian). DOI 10.46698/v3523-1431-1350-j
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