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International Theoretical Science Journal
Volume 59 >>> № 2 MARCH — APRIL 2023
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UDC 519.8
Yu.I. Kharkevych1


1 Lesya Ukrainka Volyn National University, Lutsk, Ukraine

 kharkevich.juriy@gmail.com

EXACT VALUES OF THE APPROXIMATIONS OF DIFFERENTIABLE FUNCTIONS
BY INTEGRALS OF THE POISSON TYPE

Abstract. The asymptotic properties of integrals of the Poisson type on the classes of differentiable functions are analyzed with the use of modern methods of the theory of optimal solutions and the theory of approximation of functions. Namely, the exact values of the upper bound of the deviation of the functions of the Sobolev classes from integrals of the Poisson type in the uniform metric are found. The research method used in the study makes it possible to estimate the deviation error of the classes of differentiable functions from their polyharmonic Poisson integrals with predetermined accuracy. The results obtained in the study will further contribute to the construction of higher-quality mathematical models of natural and social phenomena and therefore to more efficient solution of many problems of applied mathematics.

Keywords: polyharmonic equations, the Sobolev classes, optimization problems, asymptotic estimates, exact values of deviations.


full text

REFERENCES

  1. Chikrii A.A., Chikrii G.Ts. Matrix resolving functions in game problems of dynamics. Proc. of the Steklov Institute of Mathematics. 2015. Vol. 291. P. 56–65. https://doi.org/10.1134/S0081543815090047 .

  2. Chikrii A.A., Matychyn I.I. Game problems for fractional-order linear systems. Proc. of the Steklov Institute of Mathematics. 2010. Vol. 268. P. 54–70. https://doi.org/10.1134/S0081543810050056.

  3. Chikrii A.A., Eidelman S.D. Game problems of control for quasilinear systems with fractional Riemann-Liouville derivatives. Cybernetics and Systems Analysis. 2001. Vol. 37, N 6. P. 836–864. https://doi.org/10.1023/A:1014529914874.

  4. Vladimirov V.S. Equations of mathematical physics [in Russian]. 4th ed. Moscow: Nauka, 1981. 512 p.

  5. Zhyhallo K.M., Kharkevych Yu.I. On the approximation of functions of the Hlder class by triharmonic Poisson integrals. Ukr. Math. J. 2001. Vol. 53, N 6. P. 1012–1018. https://doi.org/10.1023/A:1013364321249 .

  6. Ka'lchuk I.V. Hrabova U.Z., Filozof L.I. Approximation of the classes harmonic Poisson integrals. J. Math. Sci. (N.Y.). 2021. Vol. 254, N 3. P. 397–405. https:// doi.org/10.1007/s10958-021-05311-8.

  7. Bushev D.N., Kharkevich Y.I. Finding solution subspaces of the Laplace and heat equations isometric to spaces of real functions, and some of their applications. Math. Notes. 2018. Vol. 103, N 5–6. P. 869–880. https://doi.org/10.1134/S0001434618050231 .

  8. Pilipenko Yu.V., Chikrij A.A. The oscillation processes of conflict control. Prikladnaya Matematika i Mekhanika. 1993. Vol. 57, N 3. P. 3–14.

  9. Chikrii A.A., Rappoport I.S. Method of resolving functions in the theory of conflict-controlled processes. Cybernetics and Systems Analysis. 2012. Vol. 48, N 4. P. 512–531. https://doi.org/10.1007/s10559-012-9430-y .

  10. Kharkevych Yu.I. On some asymptotic properties of solutions to biharmonic equations. Cybernetics and Systems Analysis. 2022. Vol. 58, N 2. P. 251–258. https://doi.org/10.1007/ s10559-022-00457-y .

  11. Chikrii A.A., Prokopovich P.V. Simple pursuit of one evader by a group. Cybernetics and Systems Analysis. 1992. Vol. 28, N 3. P. 438–444. https://doi.org/10.1007/BF01125424.

  12. Timan M.F. Approximation and properties of periodic functions [in Russian]. Kyiv: Nauk. dumka, 2009. 375 p.

  13. Kal’chuk I., Kharkevych Y. Approximation рroperties of the generalized Abel–Poisson integrals on the weyl-nagy classes. Axioms. 2022. Vol. 11, N 4. Р. 161. https://doi.org/10.3390/axioms11040161 .

  14. Abdullayev F.G., Kharkevych Yu.I. Approximation of the classes by biharmonic Poisson integrals. Ukr. Math. J. 2020. Vol. 72, N 1. P. 21–38. https://doi.org/10.1007/ s11253-020-01761-6.

  15. Stepanets A.I. Classification and approximation of periodic functions [in Russian]. Kyiv: Nauk. dumka, 1987. 268 p.

  16. Kharkevych Yu.I. On approximation of the quasi-smooth functions by their Poisson type integrals. Journal of Automation and Information Sciences. 2017. Vol. 49, N 10. P. 74–81. https://doi.org/10.1615/JAutomatInfScien.v49.i10.80.

  17. Kharkevych Yu.I. Approximative properties of the generalized Poisson integrals on the classes of functions determined by a modulus of continuity. Journal of Automation and Information Sciences. 2019. Vol. 51, N 4. P. 43–54. https://doi.org/10.1615/ JAutomatInfScien.v51.i4.40 .

  18. Zhyhallo K.M., Kharkevych Yu.I. Complete asymptotics of the deviation of a class of differentiable functions from the set of their harmonic Poisson integrals. Ukrainian Math. J. 2002. Vol. 54, N 1. P. 51–63. https://doi.org/10.1023/A:1019789402502.

  19. Kharkevych Yu.I. Asymptotic expansions of upper bounds of deviations of functions of class from their generalized Poisson integrals. Journal of Automation and Information Sciences. 2018. Vol. 50, N 8. P. 38–49. https://doi.org/10.1615/jautomatinfscien.v50.i8.40.

  20. Zhyhallo K. M., Kharkevych Yu. I. Approximation of differentiable periodic functions by their biharmonic Poisson integrals. Ukrainian Math. J. 2002. Vol. 54, N 9. P. 1462–1470. https://doi.org/10.1023/A:1023463801914 .

  21. Zhyhallo K. M., Kharkevych Yu. I. Approximation of conjugate differentiable functions by biharmonic Poisson integrals. Ukrainian Math. J. 2009. Vol. 61, N 3. P. 399–413. https://doi.org/10.1007/s11253-009-0217-x .

  22. Kal’chuk I.V., Kharkevych Y.I. Approximation of the classes by generalized Abel–Poisson integrals. Ukrainian Math. J. 2022. Vol. 74, N 9. P. 575–585. https://doi.org/10.1007/s11253-022-02084-4.

  23. Zhyhallo T.V., Kharkevych Yu.I. On approximation of functions from the class by the Abel-Poisson integrals in the integral metric. Carpathian Math. Publ. 2022. Vol. 14, N 1. P. 223–229. https://doi.org/10.15330/cmp.14.1.223-229.

  24. Kal’chuk I.V., Kharkevych Yu.I., Pozharska K.V. Asymptotics of approximation of functions by conjugate Poisson integrals. Carpathian Math. Publ. 2020. Vol. 12, N 1. P. 138–147. https://doi.org/10.15330/cmp.12.1.138-147.

  25. Timan A.F. Exact estimate of the remainder in the approximation of periodic differentiable functions by Poisson integrals. Dokl. AN SSSR. 1950. Vol. 74, N 1. P. 17–20.

  26. Zhyhallo K.M., Kharkevych Yu.I. On the approximation of functions of the Holder class by biharmonic Poisson integrals. Ukrainian Math. J. 2000. Vol. 52, N 7. P. 1113–1117. https://doi.org/10.1023/A:1005285818550.

  27. Kaniev S. On the deviation of biharmonic functions in a circle from their boundary values. Dokl. AN SSSR. 1963. Vol. 153, N 5. P. 995–998.

  28. Bushev D.M., Kharkevych Y.I. Conditions of convergence almost everywhere for the convolution of a function with delta-shaped kernel to this function. Ukrainian Math. J. 2016. Vol. 67, N 11. P. 1643–1661. https://doi.org/10.1007/s11253-016-1180-y .

  29. Kal’chuk I.V. Approximation of -differentiable functions defined on the real axis by Weierstrass operators. Ukrainian Math. J. 2007. Vol. 59, N 9. P. 1342–1363. https://doi.org/10.1007/s11253-007-0091-3 .

  30. Kharkevych Yu.I., Zhyhallo T.V. Approximation of functions from the class by Poisson biharmonic operators in the uniform metric. Ukrainian Math. J. 2008. Vol. 60, N 5. P. 769–798. https://doi.org/10.1007/s11253-008-0093-9.

  31. Chikrii A.A., Matichin I.I. Riemann–Liouville, Caputo, and sequential fractional derivatives in differential games. In: Breton M., Szajowski K. (Еds.). Advances in Dynamic Games. Annals of the International Society of Dynamic Games. Boston: Birkhuser, 2011. Vol. 11. P. 61–81. https://doi.org/10.1007/978-0-8176-8089-3_4.

  32. Chikrii A.A., Eidel’man S.D. Generalized Mittag-Leffler matrix functions in game problems for evolution equations of fractional order. Cybernetics and Systems Analysis. 2000. Vol. 36, N 3. P. 315–338. https://doi.org/10.1007/BF02732983 .

  33. Albus J., Meystel A., Chikrii A.A., Belousov A.A., Kozlov A.I. Analytical method for solution of the game problem of soft landing for moving objects. Cybernetics and Systems Analysis. 2001. Vol. 37, N 1. P. 75–91. https://doi.org/10.1023/A:1016620201241.




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