Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables
We obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Vasyl Stefanyk Precarpathian National University
2022-06-01
|
| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/5928 |
| _version_ | 1827281297889820672 |
|---|---|
| author | O.V. Fedunyk-Yaremchuk S.B. Hembars'ka |
| author_facet | O.V. Fedunyk-Yaremchuk S.B. Hembars'ka |
| author_sort | O.V. Fedunyk-Yaremchuk |
| collection | DOAJ |
| description | We obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We observe that in the multivariate case the orders of mentioned approximation characteristics of the functional classes $B^{\Omega}_{1,\theta}$ are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for the best orthogonal trigonometric approximations of the corresponding functional classes are the ordinary partial sums of their Fourier series. As a consequence of the obtained results, the exact order estimates of the orthowidths of the classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ for $d\geq2$) in the space $B_{\infty,1}$ are also established. Besides, we note that in the univariate case, in contrast to the multivariate one, the estimates of the considered approximation characteristics do not depend on the parameter $\theta$. |
| first_indexed | 2024-04-24T08:56:55Z |
| format | Article |
| id | doaj.art-b1f3e78fcc40440aba24f63808916818 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:56:55Z |
| publishDate | 2022-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-b1f3e78fcc40440aba24f638089168182024-04-16T07:10:59ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102022-06-0114117118410.15330/cmp.14.1.171-1845129Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variablesO.V. Fedunyk-Yaremchuk0S.B. Hembars'ka1Lesya Ukrainka Eastern European National University, 9 Potapova str., 43025, Lutsk, UkraineLesya Ukrainka Eastern European National University, 9 Potapova str., 43025, Lutsk, UkraineWe obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We observe that in the multivariate case the orders of mentioned approximation characteristics of the functional classes $B^{\Omega}_{1,\theta}$ are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for the best orthogonal trigonometric approximations of the corresponding functional classes are the ordinary partial sums of their Fourier series. As a consequence of the obtained results, the exact order estimates of the orthowidths of the classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ for $d\geq2$) in the space $B_{\infty,1}$ are also established. Besides, we note that in the univariate case, in contrast to the multivariate one, the estimates of the considered approximation characteristics do not depend on the parameter $\theta$.https://journals.pnu.edu.ua/index.php/cmp/article/view/5928nikol'skii-besov-type classstep hyperbolic fourier sumbest orthogonal trigonometric approximationorthowidth |
| spellingShingle | O.V. Fedunyk-Yaremchuk S.B. Hembars'ka Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables Karpatsʹkì Matematičnì Publìkacìï nikol'skii-besov-type class step hyperbolic fourier sum best orthogonal trigonometric approximation orthowidth |
| title | Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables |
| title_full | Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables |
| title_fullStr | Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables |
| title_full_unstemmed | Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables |
| title_short | Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables |
| title_sort | best orthogonal trigonometric approximations of the nikol skii besov type classes of periodic functions of one and several variables |
| topic | nikol'skii-besov-type class step hyperbolic fourier sum best orthogonal trigonometric approximation orthowidth |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/5928 |
| work_keys_str_mv | AT ovfedunykyaremchuk bestorthogonaltrigonometricapproximationsofthenikolskiibesovtypeclassesofperiodicfunctionsofoneandseveralvariables AT sbhembarska bestorthogonaltrigonometricapproximationsofthenikolskiibesovtypeclassesofperiodicfunctionsofoneandseveralvariables |