Approximation of periodic analytic functions by Fejer sums

O. O. Novikov; O. G. Rovenska

doi:10.15330/ms.47.2.196-201

For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.

1. Velichko V.E., Novikov O.A., Rovenskaya O.G., Rukasov V.I., Approximation of analytic functions by repeated de la Vallee Poussin sums, Tr. Inst. Prikl. Mat. Mekh., 22 (2011), 33–42. (in Russian)

2. Nikolskiy S.M., Approximation of the functions by trigonometric polynomials in the mean, Izv. Acad. Nauk. SSSR, Ser. Mat., 10 (1946), №3, 207–256. (in Russian)

3. Novikov O.O., Rovenska O.G., Approximation of the periodical functions of high smoothness by the rightangled Fourier sums, Carpathian Mathematical Publications, 5 (2013), №1, 111–118. (in Ukrainian)

4. Novikov O.A., Rovenska O.G., Approximation of classes of Poisson integrals by Fejer sums, Computer Research and Modeling, 7 (2015), №4, 813–819. (in Russian)

5. Novikov O.O., Rovenska O.G., Kozachenko Yu.V., Approximation of Poisson integrals by Fejer sums, Modern problems of probability theory and mathematical analysis. Scientific conference, Vorohta, 2016, 110. (in Ukrainian)

6. Rovenska O.G., Integral presentations of deviations of right-angled linear means of Fourier series on classes Cm , Bukovinsky Mat. Zh., 1 (2011), №3, 99–104. (in Ukrainian)

7. Rovenska O.O., Novikov O.O., Approximation of Poisson integrals by repeated de la Vallee Poussin sums, Nonlinear Oscillations, 13 (2010), №1, 108–111.

8. Rukasov V.I., Chaichenko, S.O., Approximation of the classes of analytical functions by de la Vallee- Poussin sums, Ukrainian Math. J., 54 (2002), №12, 2006–2024.

9. Rukasov V., Rovenska O., Integral presentations of deviations of de la Vallee Poussin right-angled sums, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math., 48 (2009), 129–137.

10. Savchuk V.V., Savchuk M.V., Chaichenko S.O., Approximation of analytic functions by de la Vallee Poussin sums, Mat. Stud., 34 (2010), №2, 207–219. (in Ukrainian)

11. Serdyuk A.S., Approximation of Poisson integrals by de la Vallee Poussin sums, Ukrainian Math. J., 56 (2004), №4, 122–134.

12. Stechkin S.B., Estimation of the remainder of Fourier series for the differentiable functions, Tr. Mat. Inst. Acad. Nauk SSSR., 145 (1980), 126–151. (in Russian)

13. Stepanec A.I., Classification and Approximation of Periodic Functions, Naukova Dumka, Kiev, 1987. (in Russian)

14. Stepanec A.I., Solution of the Kolmogorov-Nikol’skij problem for the Poisson integrals of continuous functions, Mat. Sb., 192 (2001), №1, 113–138. (in Russian)

	Approximation of periodic analytic functions by Fejer sums
Author	O. O. Novikov, O. G. Rovenska sgpi.slav@dn.ua; rovenskaya.olga.math@gmail.com Donbas State Pedagogical University, Slov’yansk, Ukraine; Donbas State Engineering Academy, Kramatorsk, Ukraine
Abstract	For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.
Keywords	asymptotic equality; analytic functions; Fejer sums; de la Vallee Poussin sums; Poisson kernel
DOI	doi:10.15330/ms.47.2.196-201
Reference	1. Velichko V.E., Novikov O.A., Rovenskaya O.G., Rukasov V.I., Approximation of analytic functions by repeated de la Vallee Poussin sums, Tr. Inst. Prikl. Mat. Mekh., 22 (2011), 33–42. (in Russian) 2. Nikolskiy S.M., Approximation of the functions by trigonometric polynomials in the mean, Izv. Acad. Nauk. SSSR, Ser. Mat., 10 (1946), №3, 207–256. (in Russian) 3. Novikov O.O., Rovenska O.G., Approximation of the periodical functions of high smoothness by the rightangled Fourier sums, Carpathian Mathematical Publications, 5 (2013), №1, 111–118. (in Ukrainian) 4. Novikov O.A., Rovenska O.G., Approximation of classes of Poisson integrals by Fejer sums, Computer Research and Modeling, 7 (2015), №4, 813–819. (in Russian) 5. Novikov O.O., Rovenska O.G., Kozachenko Yu.V., Approximation of Poisson integrals by Fejer sums, Modern problems of probability theory and mathematical analysis. Scientific conference, Vorohta, 2016, 110. (in Ukrainian) 6. Rovenska O.G., Integral presentations of deviations of right-angled linear means of Fourier series on classes Cm , Bukovinsky Mat. Zh., 1 (2011), №3, 99–104. (in Ukrainian) 7. Rovenska O.O., Novikov O.O., Approximation of Poisson integrals by repeated de la Vallee Poussin sums, Nonlinear Oscillations, 13 (2010), №1, 108–111. 8. Rukasov V.I., Chaichenko, S.O., Approximation of the classes of analytical functions by de la Vallee- Poussin sums, Ukrainian Math. J., 54 (2002), №12, 2006–2024. 9. Rukasov V., Rovenska O., Integral presentations of deviations of de la Vallee Poussin right-angled sums, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math., 48 (2009), 129–137. 10. Savchuk V.V., Savchuk M.V., Chaichenko S.O., Approximation of analytic functions by de la Vallee Poussin sums, Mat. Stud., 34 (2010), №2, 207–219. (in Ukrainian) 11. Serdyuk A.S., Approximation of Poisson integrals by de la Vallee Poussin sums, Ukrainian Math. J., 56 (2004), №4, 122–134. 12. Stechkin S.B., Estimation of the remainder of Fourier series for the differentiable functions, Tr. Mat. Inst. Acad. Nauk SSSR., 145 (1980), 126–151. (in Russian) 13. Stepanec A.I., Classification and Approximation of Periodic Functions, Naukova Dumka, Kiev, 1987. (in Russian) 14. Stepanec A.I., Solution of the Kolmogorov-Nikol’skij problem for the Poisson integrals of continuous functions, Mat. Sb., 192 (2001), №1, 113–138. (in Russian)
Pages	196-201
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
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