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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Akhiezer, N.I.: Theory of Approximation. Frederick Ungar Publishing Co., New York (1956)
Bateman, H., Erdélyi, A.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)
Kovalenko, M.D., Menshova, I.V., Shulyakovskaya, T.D.: Expansions in Fadle–Papkovich functions: examples of solutions in a half-strip. Mech. Solids 48(5), 584–602 (2013)
Kovalenko, M.D., Shulyakovskaya, T.D.: Expansion in Fadle–Papkovich functions in a strip. Theory foundations. Mech. Solids 46(5), 721–738 (2011)
Levin, B.J.: Distribution of Zeros of Entire Functions. Translated Mathematical Monographs, vol. 5. American Mathematical Society, Providence, RI (1980)
Meleshko, V.V.: Selected topics in the history of two-dimensional biharmonic problem. Appl. Mech. Rev. 56(1), 33–85 (2003)
Paley, R., Wiener, N.: Fourier Transforms in the Complex Domain. American Mathematical Society, New York (1934)
Pflüger, A.: Über Eine Interpretation Gewisser Konvergenz- und Fortsetzungseigenschaften Dirichletscher Reihen. Comment. Math. Helv. 8, 89–129 (1935/36)
Vladimirov, V.S.: Equations of Mathematical Physics. M. Dekker, New York (1971)
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.
Communicated by Fabrizio Colombo.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s11785-017-0643-y
- Entire function
- Quasi-entire function
- Borel transform
Mathematics Subject Classification