Cite this article as:

Shishkin A. B. Factorization of Entire Symmetrical Functions of Exponential Type. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 1, pp. 42-68. DOI:


Factorization of Entire Symmetrical Functions of Exponential Type


Let π be an entire function of minimal type of order 1. The entire function F is called π-symmetric if it is represented in the form of a composition f ◦ π, where the f is an entire function. The article deals with the following question. Can we present every π-symmetric function of exponential type as a product of two functions with a close growth, each of which is itself an entire π-symmetric function? This question is answered in the affirmative, but under certain restrictions on for the subordinate function π. For example, an entire function of completely regular growth at proximate order ρ(r) ≈ ρ ∈ (0;1) with constant positive indicator is subject to these restrictions. Other examples relate to the reversibility of the entire function in the circles of constant radius whose centers lie outside some exceptional set.


1. Ehrenpreis L. Solution of some problems division. IV. Amer. J. Math., 1960, vol. 82, pp. 522–588.

2. Dicson D. G. Factoring Fourier transforms with zeros in a strip. Proc. Amer. Math. Soc., 1989, vol. 106, pp. 107–114.

3. Yulmukhametov R. S. Solution of the Ehrenpreis factorization problem. Sb. Math., 1999, vol. 190, no. 4, pp. 597–629. DOI:

4. Shishkin A. B. Projective and injective descriptions in the complex domain. Duality. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 47–64 (in Russian).

5. Volkovaya T. A., Shishkin A. B. Lokal’noe opisanie tselykh funktsii [Local description of entire functions]. Issledovaniia po matematicheskomu analizu. Itogi nauki. Iug Rossii. Mat. forum [Research on mathematical analysis. The results of science. South of Russia. Mat. forum]. Vladikavkaz, UMI VSC RAS, 2014, vol. 8, pt. 1, pp. 218–230 (in Russian).

6. Volkovaya T. A., Shishkin A. B. Lokal’noe opisanie tselykh funktsii. Podmoduli ranga 1 [Local description of entire functions. Submodules of rank 1]. Vladikavkaz. Mat. Zh., 2014, vol. 16, no. 2, pp. 14–28 (in Russian).

7. Krasichkov-Ternovskii I. F. Invariant subspaces of analytic functions. I. Spectral analysis on convex regions. Math. USSR-Sb., 1972, vol. 16, no. 4, pp. 471–500.

8. Azarin V. S. On the decomposition of an entire function of finite order into factors having given growth. Math. USSR-Sb., 1973, vol. 19, no. 2, pp. 225–226.

9. Krasichkov-Ternovskii I. F. Spectral synthesis in a complex domain for a differential operator with constant coefficients. IV : Synthesis. Russian Acad. Sci. Sb. Math., 1993, vol. 76, no. 2, pp. 407–426.

10. Khabibullin B. N. Teoremy sravneniia i odnorodnosti dlia subgarmonicheskikh funktsii. Dis. ... kand. fiz.-mat. nauk [Comparison theorems for subharmonic functions : Dr. phys. and math. sci. diss.]. Rostov-on-Don, 1985. 103 p. (in Russian).

11. Krasichkov I. F. Sravnenie tselykh funktsii tselogo poriadka po raspredeleniiu ikh kornei [Comparison of entire functions of finite order by means of the distribution of their roots]. Mat. Sb., 1966, vol. 70(112), no. 2, pp. 198–230 (in Russian).

12. Pis’mennyi R. G. Factoring of an entire function into two equivalent functions. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2009, vol. 9, iss. 1, pp. 19–30 (in Russian).

13. Pis’mennyi R. G., Shishkin A. B. Rasshcheplenie tselykh funktsii konechnogo poriadka na ekvivalentnye mnozhiteli [Splitting entire functions of finite order in the equivalent factors]. Vest. Adyg. gos. un-ta. Ser. Estestvennomatematicheskie i tekhnicheskie nauki, 2010, no. 2(61), pp. 23–28 (in Russian).

14. Volkovaya T. A. Synthesis in the Polynomial Kernel of Two Analytic Functionals. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 3, pp. 251–262 (in Russian).

15. Leont’ev A. F. Riady eksponent [Exponential series]. Moscow, Nauka, 1976, 536 p. (in Russian).

16. Gol’dberg A. A., Ostrovskii I. V. O proizvodnykh i pervoobraznykh tselykh funktsii vpolne reguliarnogo rosta [Derivatives and primitives of entire functions of completely regular growth]. Teoriia funktsii, funkts. analiz i ikh pril. (Kharkiv), 1973, no. 18, pp. 70–81 (in Russian).

Full text: