Integral representation of one class of entire functions


  • Ruslan Khats' Drohobych Ivan Franko State Pedagogical University



Paley-Wiener theorem, entire function of exponential type, Ordinary Differential Equations, Schwarz inequality, asymptotic estimate


In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class.


M.M. Dzhrbashyan, Integral transforms and representations of functions in the complex domain, Nauka, Moscow, 1966. (in Russian)

I. Laine, Nevanlinna theory and complex differential equations, Walter de Gruiter, Berlin, 2011.

B.Ya. Levin, Lectures on entire functions. Transl. Math. Monogr., Amer. Math. Soc., Providence, R.I., 150, 1996.

V.I. Lutsenko and R.S. Yulmukhametov, Generalization of the Paley-Wiener theorem in weighted spaces, Math. Notes, 48 (1990), no. 5, pp. 1131-1136.

A.M. Sedletskii, Analytic Fourier transforms and exponential approximations. I., J. Math. Sci., 129 (2005), no. 6, pp. 4251-4408.

V.K. Tuan and A.I. Zayed, Paley-Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl., 266 (2002), no. 1, pp. 200-226.

B.V. Vynnyts'kyi and V.M. Dilnyi, On approximation properties of one trigonometric system, Russ. Math., 58 (2014), no. 11, pp. 10-21.

B.V. Vynnyts'kyi and R.V. Khats', Some approximation properties of the systems of Bessel functions of index $-3/2$, Mat. Stud., 34 (2010), no. 2, pp. 152-159.

B.V. Vynnyts'kyi and R.V. Khats', Completeness and minimality of systems of Bessel functions, Ufa Math. J., 5 (2013), no. 2, pp. 131-141.

B.V. Vynnyts'kyi and R.V. Khats', On the completeness and minimality of sets of Bessel functions in weighted $L^2$-spaces, Eurasian Math. J., 6 (2015), no. 1, pp. 123-131.

B.V. Vynnyts'kyi and O.V. Shavala, Boundedness of solutions of a second-order linear differential equation and a boundary value problem for Bessel's equation, Mat. Stud., 30 (2008), no. 1, pp. 31-41. (in Ukrainian)

B.V. Vynnyts'kyi and O.V. Shavala, Some properties of boundary value problems for Bessel's equation, Math. Bull. Shevchenko Sci. Soc., 10 (2013), pp. 169-172.

G.N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.

N. Wiener and R.C. Paley, Fourier transforms in the complex domain, USA: Amer. Math. Soc., Providence, R.I., 19, 1934.




How to Cite

Khats’, R. (2022). Integral representation of one class of entire functions. Armenian Journal of Mathematics, 14(1), 1–9.