Asymptotics of approximation of functions by conjugate Poisson integrals
Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipsch...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2020-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/3893 |
| _version_ | 1827281270139256832 |
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| author | I.V. Kal'chuk Yu.I. Kharkevych K.V. Pozharska |
| author_facet | I.V. Kal'chuk Yu.I. Kharkevych K.V. Pozharska |
| author_sort | I.V. Kal'chuk |
| collection | DOAJ |
| description | Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness. |
| first_indexed | 2024-04-24T08:57:23Z |
| format | Article |
| id | doaj.art-45ea94f144854341b80af949c895d080 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:23Z |
| publishDate | 2020-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-45ea94f144854341b80af949c895d0802024-04-16T07:00:37ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102020-06-0112113814710.15330/cmp.12.1.138-1473386Asymptotics of approximation of functions by conjugate Poisson integralsI.V. Kal'chuk0https://orcid.org/0000-0001-8822-3716Yu.I. Kharkevych1https://orcid.org/0000-0002-8577-5096K.V. Pozharska2https://orcid.org/0000-0001-7599-8117Lesya Ukrainka East European National University, 13 Voli avenue, 43025, Lutsk, UkraineLesya Ukrainka East European National University, 13 Voli avenue, 43025, Lutsk, UkraineInstitute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, UkraineAmong the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class $\textrm{Lip}_1\alpha $, i.e. the class of continuous $ 2\pi $-periodic functions satisfying the Lipschitz condition of order $\alpha$, $0<\alpha\le 1$, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymptotic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class $\textrm{Lip}_1\alpha $. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.https://journals.pnu.edu.ua/index.php/cmp/article/view/3893poisson integralasymptotic expansionconjugate functionkolmogorov-nikol'skii problem |
| spellingShingle | I.V. Kal'chuk Yu.I. Kharkevych K.V. Pozharska Asymptotics of approximation of functions by conjugate Poisson integrals Karpatsʹkì Matematičnì Publìkacìï poisson integral asymptotic expansion conjugate function kolmogorov-nikol'skii problem |
| title | Asymptotics of approximation of functions by conjugate Poisson integrals |
| title_full | Asymptotics of approximation of functions by conjugate Poisson integrals |
| title_fullStr | Asymptotics of approximation of functions by conjugate Poisson integrals |
| title_full_unstemmed | Asymptotics of approximation of functions by conjugate Poisson integrals |
| title_short | Asymptotics of approximation of functions by conjugate Poisson integrals |
| title_sort | asymptotics of approximation of functions by conjugate poisson integrals |
| topic | poisson integral asymptotic expansion conjugate function kolmogorov-nikol'skii problem |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/3893 |
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