On interpolation problem with derivative in the space of
entire functions with fast-growing interpolation knots

B. V. Vynnyts’kyi; I. B. Sheparovych

doi:10.15330/ms.51.1.50-58

We obtaine the conditions on a sequence $(b_{k,1}; b_{k,2}),\ k \in {\mathbb N},$ such that the interpolation problem $g(\lambda _{k} ) = b_{k,1} ,\ {g}'(\lambda _{k} ) = b_{k,2} $ has the unique solution in a subspace of entire functions $ g $ satisfying the condition $\ln M_g (r)\le c_1\exp\left(N(r)+N(\rho_1 r)\right)$, where $\vert \lambda_{k}/\lambda_{k + 1}\vert \le \Delta <1$, and $N(r)$ is the Nevanlinna counting function of the sequence ($\lambda_k$). These results have been applied to describe solutions of the differential equation $f''+a_0 f=0$ with a coefficient $a_0$ from the some space of entire functions.

1. A.O. Gelfond, Calculus of finite differences, Nauka, Moscow, 1967, Russian; English translation: Hindustan Publishing, Delhi, 1971.

2. Yu.A. Kazmin, On some Gelfonds problem, Mat. sb., 132 (1973), №4, 520-543. (in Russian)

3. B.Ja. Levin, Distribution of zeros of entire functions, GITTL, Moscow, 1956, Russian; English transl., Amer. Math. Soc., Providence, R. I., 1964.

4. A.F. Leontev, Entire function. Series of exponent, Nauka, Moscow, 1983. (in Russian)

5. G.P. Lapin, Interpolation in the class of entire functions of finite order, Izv. Vyssh. Uchebn. zaved. Mat., 5 (1959), №12, 146-153. (in Russian)

6. A.V. Bratishchev, On the interpolation problem in the some classes of entire functions, Sib. mat., 2 (1976), №17, 30-44. (in Russian)

7. Ju.F. Korobeinik, On the some interpolation problem for entire functions, Izv. Vyssh. Uchebn. zaved. Mat., 2 (1985), 37-45. (in Russian)

8. O.A. Bozhenko, A.F. Grishyn, K.G. Maljutin, Interpolation in the class of entire functions of zero order, Mat. sb., 79 (2015), №2, 3-28. (in Russian)

9. A.F. Grishyn, A.M. Russakovskyi, Free interpolation by entire functions, Teoriya funkc., funkc. Analiz i ikh prilozh, 44, 1985, 32-42. (in Russian)

10. B.V. Vynnytskyi, On the representation of entire functions by series ${\sum\nolimits_{n = 1}^{\infty} {d_{n} f(\lambda _{n}z)}}$, Mat. zamet., 27 (1980), №3, 361-372. (in Russian)

11. B.V. Vynnytskyi, On the representation of entire functions by series ${\sum\nolimits_{n = 1}^{\infty} {d_{n} f(\lambda _{n}z)}}$, Mat. zamet., 29 (1981), №4, 503-516. (in Russian)

12. B.V. Vynnytskyi, I.B. Sheparovych, On the interpolation sequences of some functions class, analytic in the unit disk, Ukr. mat., 53 (2001), №7, 879-886, (in Ukrainian)

13. B.V. Vynnytskyi, I.B. Sheparovych, On the existence of solution of interpolation problem in the some entire functions class that is determinate by account function, Aktualni problemi fiziki, matematiki ta informatiki, 4 (2012), 12-14. (in Ukrainian)

14. B.V. Vynnytskyi, On the construction of an entire function of arbitrary order with given asymptotic properties, Ukr. Mat., 38 (1986), №2, 143-148. (in Russian)

15. B.V. Vynnytskyi, On the description of the bases of generalized exponential systems, Mat. sb., 135 (177) (1988), №1, 59-79. (in Russian)

16. I.I. Ibragimov, Interpolation methods and some of their applications, Nauka, Moscow, 1971, 520. (in Russian)

17. V. Seda, On some properties of solutions of the differential equation ${y}'' = Q(z)y$, where $Q(z)\not{\equiv} 0$ is an entire function, Acta F.R.N. Univ. Comen. Mathem., 4, 1959, 223-253. (in Slovak)

18. J. Heittokangas, I. Laine, Solutions of ${f}'' + A(z)f = 0$ with prescribed sequences of zeros, Acta Math. Univ. Comenianae, 74 (2005), №2, 287-307.

19. S. Bank, A note on the zero.sequences of solutions of linear differential equations, Results in Mathematics, 13 (1988), 1-10.

20. J. Grohn, J. Heittokangas, New findings on Bank-Sauer approach in oscillatory theory, Constr. Approx., 35 (2012), 345-361.

21. G.G. Gundersen, Estimates for the logarithmic derivative of a meromophic function plus similar estimates, J. Lond. Math. Soc., (1) 37 (1988), 88-104.

22. S. Bank, A general theorem concerning the growth of solution of first-order algebraic differential equations, Compos. Math., 25 (1972), 61-70.

23. B. Vynnytskyi, O. Shavala, Remark on Seda theorem, Acta Math. Univ. Comenianae, 81 (2012), №1, 55-60.

24. I. Chyzhykov, I. Sheparovych, Interpolation of analytic functions of moderate growth in the unit disk and zeros of solutions of a linear differential equation, J. Math. Anal. Appl., 414 (2014), №1, 319-333. doi: 10.1016/j.jmaa.2013.12.066

	On interpolation problem with derivative in a space of entire functions with fast-growing interpolation knots
Author	B. V. Vynnyts’kyi¹, I. B. Sheparovych² vynnytskyi@ukr.net¹, isheparovych@ukr.net² Ivan Franko State Pedagogical University of Drohobych, Department of Physics, Mathematics, Economy And Innovative Technologies, Drohobych, Ukraine
Abstract	We obtaine the conditions on a sequence $(b_{k,1}; b_{k,2}),\ k \in {\mathbb N},$ such that the interpolation problem $g(\lambda _{k} ) = b_{k,1} ,\ {g}'(\lambda _{k} ) = b_{k,2} $ has the unique solution in a subspace of entire functions $ g $ satisfying the condition $\ln M_g (r)\le c_1\exp\left(N(r)+N(\rho_1 r)\right)$, where $\vert \lambda_{k}/\lambda_{k + 1}\vert \le \Delta <1$, and $N(r)$ is the Nevanlinna counting function of the sequence ($\lambda_k$). These results have been applied to describe solutions of the differential equation $f''+a_0 f=0$ with a coefficient $a_0$ from the some space of entire functions.
Keywords	interpolation problem; entire function; interpolation knots; differential equation
DOI	doi:10.15330/ms.51.1.50-58
Reference	1. A.O. Gelfond, Calculus of finite differences, Nauka, Moscow, 1967, Russian; English translation: Hindustan Publishing, Delhi, 1971. 2. Yu.A. Kazmin, On some Gelfonds problem, Mat. sb., 132 (1973), №4, 520-543. (in Russian) 3. B.Ja. Levin, Distribution of zeros of entire functions, GITTL, Moscow, 1956, Russian; English transl., Amer. Math. Soc., Providence, R. I., 1964. 4. A.F. Leontev, Entire function. Series of exponent, Nauka, Moscow, 1983. (in Russian) 5. G.P. Lapin, Interpolation in the class of entire functions of finite order, Izv. Vyssh. Uchebn. zaved. Mat., 5 (1959), №12, 146-153. (in Russian) 6. A.V. Bratishchev, On the interpolation problem in the some classes of entire functions, Sib. mat., 2 (1976), №17, 30-44. (in Russian) 7. Ju.F. Korobeinik, On the some interpolation problem for entire functions, Izv. Vyssh. Uchebn. zaved. Mat., 2 (1985), 37-45. (in Russian) 8. O.A. Bozhenko, A.F. Grishyn, K.G. Maljutin, Interpolation in the class of entire functions of zero order, Mat. sb., 79 (2015), №2, 3-28. (in Russian) 9. A.F. Grishyn, A.M. Russakovskyi, Free interpolation by entire functions, Teoriya funkc., funkc. Analiz i ikh prilozh, 44, 1985, 32-42. (in Russian) 10. B.V. Vynnytskyi, On the representation of entire functions by series ${\sum\nolimits_{n = 1}^{\infty} {d_{n} f(\lambda _{n}z)}}$, Mat. zamet., 27 (1980), №3, 361-372. (in Russian) 11. B.V. Vynnytskyi, On the representation of entire functions by series ${\sum\nolimits_{n = 1}^{\infty} {d_{n} f(\lambda _{n}z)}}$, Mat. zamet., 29 (1981), №4, 503-516. (in Russian) 12. B.V. Vynnytskyi, I.B. Sheparovych, On the interpolation sequences of some functions class, analytic in the unit disk, Ukr. mat., 53 (2001), №7, 879-886, (in Ukrainian) 13. B.V. Vynnytskyi, I.B. Sheparovych, On the existence of solution of interpolation problem in the some entire functions class that is determinate by account function, Aktualni problemi fiziki, matematiki ta informatiki, 4 (2012), 12-14. (in Ukrainian) 14. B.V. Vynnytskyi, On the construction of an entire function of arbitrary order with given asymptotic properties, Ukr. Mat., 38 (1986), №2, 143-148. (in Russian) 15. B.V. Vynnytskyi, On the description of the bases of generalized exponential systems, Mat. sb., 135 (177) (1988), №1, 59-79. (in Russian) 16. I.I. Ibragimov, Interpolation methods and some of their applications, Nauka, Moscow, 1971, 520. (in Russian) 17. V. Seda, On some properties of solutions of the differential equation ${y}'' = Q(z)y$, where $Q(z)\not{\equiv} 0$ is an entire function, Acta F.R.N. Univ. Comen. Mathem., 4, 1959, 223-253. (in Slovak) 18. J. Heittokangas, I. Laine, Solutions of ${f}'' + A(z)f = 0$ with prescribed sequences of zeros, Acta Math. Univ. Comenianae, 74 (2005), №2, 287-307. 19. S. Bank, A note on the zero.sequences of solutions of linear differential equations, Results in Mathematics, 13 (1988), 1-10. 20. J. Grohn, J. Heittokangas, New findings on Bank-Sauer approach in oscillatory theory, Constr. Approx., 35 (2012), 345-361. 21. G.G. Gundersen, Estimates for the logarithmic derivative of a meromophic function plus similar estimates, J. Lond. Math. Soc., (1) 37 (1988), 88-104. 22. S. Bank, A general theorem concerning the growth of solution of first-order algebraic differential equations, Compos. Math., 25 (1972), 61-70. 23. B. Vynnytskyi, O. Shavala, Remark on Seda theorem, Acta Math. Univ. Comenianae, 81 (2012), №1, 55-60. 24. I. Chyzhykov, I. Sheparovych, Interpolation of analytic functions of moderate growth in the unit disk and zeros of solutions of a linear differential equation, J. Math. Anal. Appl., 414 (2014), №1, 319-333. doi: 10.1016/j.jmaa.2013.12.066
Pages	50-58
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On interpolation problem with derivative in a space of entire functions with fast-growing interpolation knots