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Borel Transform in the Class W of Quasi-entire Functions


In this article the class W of quasi-entire functions is introduced and the properties of the Borel transform in this class are studied. The class W includes the quasi-entire functions of exponential type whose module is square-integrable on the whole real axis. For such functions the Borel transform formulae can be represented in the form of integrals of Cauchy type from the jumps of functions that are analytic in the whole complex plane outside some cuts in which the jumps are defined. Due to this fact it is possible to obtain Parseval type equalities in the considered class of quasi-entire functions. These results, as a consequence, lead to the known Paley–Wiener results (in particular, the Paley–Wiener theorem). Therefore, they can be considered as a generalization of similar results for the class W of entire functions of exponential type introduced by them. There are different examples of the Borel transform in the class W of quasi-entire functions. In particular, the examples are chosen to show how the obtained results can be generalized in case the module of the quasi-entire function has at most power growth on the real axis. A similar generalization in the class of entire functions of exponential type is known as the Paley–Wiener–Schwartz theorem. In conclusion, there are simple generalizations for the case when the cut which ensures the single-valuedness of the function Borel-associated with the quasi-entire function of the class W does not necessarily coincide with the real axis.

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This research was supported by the Russian Foundation for Basic Research, Research Project Nos. 16-31-60028 mol_a_dk and 15-41-02644 r_povolzhe_a.

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Correspondence to Irina V. Menshova.

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Communicated by Fabrizio Colombo.

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Kerzhaev, A.P., Kovalenko, M.D. & Menshova, I.V. Borel Transform in the Class W of Quasi-entire Functions. Complex Anal. Oper. Theory 12, 571–587 (2018).

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  • Entire function
  • Quasi-entire function
  • Borel transform

Mathematics Subject Classification

  • 30D60