Multiple interpolation with the fast-growing knots in the class of entire functions and its application

Document Type : Research Article

Author

• I. Sheparovych

Department of Physics, Mathematics, Economy and Innovative Technologies, Ivan Franko State Pedagogical Institute of Drohobych, Drohobych, Ukraine.

10.22067/ijnao.2021.70074.1028

Abstract

The conditions for the sequence of complex numbers (bn,k) are obtained, such that the interpolation problem g(k-1)(λn) = bn,k, k ∈ 1, s, n ∈ N, where |λk/λk+1| ≤ ∆ < 1, has a unique solution in some classes of entire functions g for which Mg(r) ≤ c1 exp ((s - 1)N(r) + N(ρ1r)), where N(r) is the counting function of the sequence (λn), ρ1 ∈ (∆; 1), and c1 > 0. Moreover, these results have been applied to the description of the solution of the differential equation f(s) +A0(z)f = 0 for which (λn) is zero-sequence and the coeffcient A0 is an entire function from the mentioned class.

Keywords

• Interpolation problem
• Entire function
• Solution of differential equation

20.1001.1.24236977.2022.12.1.6.0

Main Subjects

• Numerical Analysis

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References

1. Abanina, T.I.Interpolation problem in spaces of entire functions of arbitrarily rapid growth, Vinniti no. 5440–5488, Moscow, 1988 (in Russian).
2. Bank S. A note on the zero–sequences of solutions of linear differential equations Results in Mathematics, 13 (1988), 1–10.
3. Bank S. and Laine I. On the oscillation theory of f′′ + A(z)f = 0 where A is entire Trans. Am. Math. Soc., 273(1) (1982), 287–307.
4. Bozhenko, O.A. and Maljutin, K.G. The multiple interpolation problem in the class of entire functions of zero order, Proceedings of the Institute of Mathematics of NAS of Ukraine, 10 (4-5) (2013), 412–423 (in Russian).
5. Bratishchev, A.V. The interpolation problem in certain classes of entire functions Sib. Mat. J. 17(1) (1976), 30–40 (in Russian).
6. Bratishchev, A.V. and Korobeinik, Yu.F. The multiple interpolation problem in the space of entire functions of given proximate order Izv. Akad. Nauk SSSR, Ser. Mat., 40(5) (1976), 1102–1127 (in Russian).
7. Cao T.-B. and Yi, H.-X. The growth of solutions of linear differential equations with coeffcients of iterated order in the unit disc, J. Math. Anal. Appl., 319 (2006), 1, 278–294.
8. Chyzhykov, I. and Long J. On non-separated zero sequences of solutions of a linear differential equation, Proc. Edinburgh Math. Soc., 64 (2021), 247–261.
9. Chyzhykov, I. and Sheparovych, I. Interpolation of analytics functions of moderate growth in the unit disk and zeros of solutions of a linear differential equation, J. Math. Analysis and Applications, 2014, 319–333.
10. Firsakova, O.S.Some problems connected with interpolation by means of entire functions, Mathematics of the USSR-Izvestiya 120(3) (1958), 447–480 (in Russian).
11. Gel’fond, A. O. Calculus of finite differences Nauka, Moscow, 1967, Russian. English translation: Hindustan Publishing, Delhi, 1971.
12. Gröhn, J. and Heittokangas, J. New findings on Bank-Sauer approach in oscillatory theory, Constr. Approx., 35 (2012), 345–361.
13. Heittokangas J. and Laine I. Solutions of f′′ + A(z)f = 0 with prescribed sequences of zeros Acta Mathematica Universitatis Comenianae, 74(2) (2005), 287–307.
14. Kaz’min Yu, A. On some Gel’fond’s problem, Mat. sb. 132(4), 1973, 520–543 (in Russian).

15. Lapin, G.P. Interpolation in the class of entire functions of finite order Izv. vys. uch. zavedenij. Matematika, 12(5) (1959), 146–153 (in Russian).
16. Lapin, G.P. On entire functions of finite order which assume, together with their derivatives given values at given points Sib. Mat. J. 6(6) (1965), 1267–1281 (in Russian).
17. Tu J. and Long, T. Oscillation of complex high order linear differential equations with coeffcients of finite iterated order, Electron. J. Qual. Theory Differ. Equ., 66 (2009), 1–13.
18. Vynnyts’kyi, B.V. On the representation of entire functions by series P∞ n=1 dnf(λnz) Mat. zamet., 27(3) (1980), 361–372 (in Russian).
19. Vynnyts’kyi, B.V. On the representation of entire functions by series P∞ n=1 dnf(λnz) Mat. zamet., 29(4) (1981), 503–516 (in Russian).
20. Vynnyts’kyi, B.V. On the construction of an entire function of arbitrary order with given asymptotic properties, Ukr. Mat. J., 38(2) (1986), 143–148 (in Russian).
21. Vynnyts’kyi, B.V. and Sheparovych I.B. On interpolation problem with derivative in the space of entire functions with fast-growing interpolation knots Mathem. Studii, 51(1) (2019) 50–58.