Abstract—
Let \(Z\) and \(W\) be two distributions of points on the complex plane \(\mathbb{C}\). In the case of \(Z\) and \(W\), lying on the positive half-line \({{\mathbb{R}}^{ + }} \subset \mathbb{C}\), the classic Malliavin–Rubel theorem from the 1960s, gives a necessary and sufficient correlation between \(Z\) and \(W\), when for each entire function \(g \ne 0\) exponential type that vanishes on \(W\), there exists an entire function \(f \ne 0\) exponential type that vanishes on \(Z\), with the constraint \({\text{|}}f{\text{|}} \leqslant {\text{|}}g{\text{|}}\) on the imaginary axis \(i\mathbb{R}\). In subsequent years, this theorem was extended to \(Z\) and \(W\), located outside of some pair of angles containing \(i\mathbb{R}\) inside. Our version of the Malliavin–Rubel theorem admits the location of \(Z\) and \(W\) near and on \(i\mathbb{R}\).
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ACKNOWLEDGMENTS
I am profoundly grateful to a reviewer for noting a number of inaccuracies in the original version of this paper and kindly correcting them.
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This investigation was performed with financial support from the Russian Foundation for Basic Research as part of the postgraduate scientific project no. 20-31-90074.
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Translated by A. Muravnik
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Salimova, A.E. A Version of the Malliavin–Rubel Theorem on Entire Functions of Exponential Type with Zeros near the Imaginary Axis. Russ Math. 66, 37–45 (2022). https://doi.org/10.3103/S1066369X22080072
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DOI: https://doi.org/10.3103/S1066369X22080072