Kuryliak A. Asymptotic properties and value distribution of random analytic functions.

Українська версія

Thesis for the degree of Doctor of Science (DSc)

State registration number

0524U000127

Thesis Registration Form

0524U000127.pdf

Applicant for

Specialization

  • 01.01.01 - Математичний аналіз

16-05-2024

Specialized Academic Board

Д 20.051.09

Vasyl Stefanyk Precarpathian National University

Essay

Object of study: analytical and random analytical functions of one and many variables, the domain of convergence of which can be any multiple-circular Reinhardt domain, lacunar series of homogeneous polynomials, integer multiple Dirichlet series, Laplace-Stilts integrals. Research methods: methods of function theory, multidimensional complex analysis, probability theory, as well as certain techniques from the works of A. A. Goldberg, M. M. Sheremeta, O. B. Skaskiv, M. L. Sodin and their students are used. Scientific novelty of the results. All the results of the dissertation submitted for defence are new. The following results have been obtained for the first time in this thesis: 1) analogues of the Borel relation and the Wiener inequality for analytical functions of one variable that can be represented as a power series with a radius of convergence R ∈ (0; +∞] are obtained; 2) the existence of the Levy effect for integers and analytical functions in the range was verified in the case when the sequence of random variables that are multipliers of Taylor coefficients of a random analytical function may not be uniformly bounded; 3) the Wiman-type inequality for integer functions of many complex variables is refined, an example is constructed to verify the accuracy of the obtained inequality and the presence of the Levy effect is checked; 4) exact analogues of the Bitlian-Goldberg inequality are obtained for integer functions of many complex variables defined by lacunar series in terms of homogeneous polynomials; 5) analogues of Wiman type inequality are established and the existence of the Levy effect is checked for analytical functions with convergence domains (a) C p ; (b) D l × C p-l ; (c) D p ; where l, p ∈ N, p > l, p ≥ 2, and examples are constructed to prove their accuracy in each of these sets; 6) analogues of the Wiener inequality for analytical functions in any multiple circular Reinhardt domain are obtained, and the existence of the Levy effect for these functions is verified; 7) exact analogues of the Wiener inequality for integer multiples of Dirichlet series with arbitrary complex exponents are proved; 8) top and bottom bounds for the probability of zero absence for random integer functions and some analytical functions were obtained and examples were constructed to prove their accuracy; 9) exact Borel-type relations for Laplace-Stilles integrals were established; 10) the properties of the Banach space of Laplace-Stieltjes integrals and Dirichlet series are investigated; 11) statements about generalised and modified generalised orders of Laplace-Stieltjes integrals are obtained; 12) the properties of the Fr´echet space of integer Dirichlet series of finite generalised order are investigated.

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Research papers

1.Kuryliak A.O., Skaskiv O.B., Zikrach D.Yu. On Borel’s type relation for the Laplace–Stieltjes integrals. Mat. Stud. 2014. V. 42. № 2. P. 134–142.

2. Kuryliak A.O., Skaskiv O.B., Stasiv N.Yu. On the convergence of Dirichlet series with random exponents. Int. J. Appl. Math. 2017. V. 30. № 3. P. 229–238.

3. Kuryliak A. Subnormal independent random variables and Levy’s phenomenon for entire functions. Mat. Stud. 2017. V. 47. № 1. P. 10–19.

4. Sheremeta M.M., Dobushovskyy M.S., Kuryliak A.O. On a Banach space of Laplace-Stieltjes integrals. Mat. Stud. 2017. V. 48. № 2. P. 143–149.

5. Kuryliak A.O., Tsvigun V.L. Wiman’s type inequality for multiple power series in an unbounded cylinder domain. Mat. Stud. 2018. V. 49. № 1. P. 29–51.

6. Kuryliak A.O., Skaskiv O.B., Stasiv N.Yu. On the convergence of random multiple Dirichlet series. Mat. Stud. 2018. V. 49. № 2. P. 122–137.

7. Kuryliak A.O., Tsvigun V.L. Wiman’s inequality for analytic functions in D×C with rapidly oscillating coefficients. Carpathian Math. Publ. 2018. V. 10. № 1. P. 133–142.

8. Sheremeta M.M., Kuryliak A.O. On the growth of Laplace-Stieltjes integrals. Mat. Stud. 2018. V. 50. № 1. P. 22–35.

9. Kuryliak A., Skaskiv O., Skaskiv S. Levy’s phenomenon for analytic functions in the polydisc. Eur. J. Math. 2020. V. 6. P. 138–152.

10. Kuryliak A.O., Panchuk S.I., Skaskiv O.B. Bitlyan-Gol’dberg type inequality for entire functions and diagonal maximal term. Mat. Stud. 2020. V. 54. № 2. P. 135–145.

11. Kuryliak A.O., Skaskiv O.B. Wiman’s type inequality for analytic and entire functions and h-measure of an exceptional sets. Carpathian Math. Publ. V.

12. 2020. № 2. P. 492–498. 12. Kuryliak A.O., Skaskiv O.B. Wiman’s type inequality for some double power series. Bukovinian Math. J. 2021. V. 9. № 1. P. 56–63.

13. Kuryliak A.O., Skaskiv O.B. Wiman’s type inequality in multiple-circular domain. Axioms. 2021. V. 10. № 4. 348.

14. Kuryliak A.O., Skaskiv O.B. Wiman-type inequality in a multiple-circular domain: L´evy’s phenomenon and exceptional sets. Ukrainian Math. J. 2022. V. 74. № 5. P. 743–756.

15. Куриляк А., Скаскiв О. Нерiвнiсть типу Вiмана для степеневих рядiв з швидко коливними коефiцiєнтами в кратно-кругових областях. Вiсник Львiв. ун-ту. Сер. мех.-мат. 2022. Т. 93. P. 83–96.

16. Kuryliak A.O., Skaskiv O.B. Entire Gaussian functions: probability of zeros absence. Axioms. 2023. V. 12. № 3. 255.

17. Kuryliak A.O., Skaskiv O.B. Analytic Gaussian functions in the unit disc: probability of zeros absence. Mat. Stud. 2023. V. 59. № 1. 29–45.

18. Куриляк А.О., Шеремета М.М. Про простори Банаха i Фреше iнтегралiв Лапласа–Стiлтьєса. Нелiнiйнi коливання. Т. 24. № 2. С. 185–196 (2021). Engl. transl.: Kuryliak A.O., Sheremeta M.M. On Banach spaces and Frechet spaces of Laplace–Stieltjes integrals. J. Math. Sci. (US). 2023. V. 270. № 2. P. 280–293.

19. Kuryliak A.O. Wiman’s type inequality for entire multiple Dirichlet series with arbitrary complex exponents. Mat. Stud. 2023. V. 59. № 2. P. 178–186.

20. Kuryliak A.O., Skaskiv O.B. Sub-Gaussian random variables and Wiman’s inequality. Carpathian Math. Publ. 2023. V. 15. № 1. P. 306–314.

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Files

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