Non-symmetric approximations of functional classes by splines on the real line
Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2021-12-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/5570 |
| _version_ | 1797205794298527744 |
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| author | N.V. Parfinovych |
| author_facet | N.V. Parfinovych |
| author_sort | N.V. Parfinovych |
| collection | DOAJ |
| description | Let $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$.
We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$.
These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency.
As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$. |
| first_indexed | 2024-04-24T08:56:47Z |
| format | Article |
| id | doaj.art-7f72bb8dbe0a4636b71ed38b36b6e00b |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:56:47Z |
| publishDate | 2021-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-7f72bb8dbe0a4636b71ed38b36b6e00b2024-04-16T07:09:07ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102021-12-0113383183710.15330/cmp.13.3.831-8374813Non-symmetric approximations of functional classes by splines on the real lineN.V. Parfinovych0https://orcid.org/0000-0002-3448-3798Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, UkraineLet $S_{h,m}$, $h>0$, $m\in {\mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $k\in {\mathbb Z}$. We obtain exact values of the best $(\alpha, \beta)$-approximations by spaces $S_{h,m}\cap L_1({\mathbb R})$ in the space $L_1({\mathbb R})$ for the classes $W^r_{1,1}({\mathbb R})$, $r\in {\mathbb N}$, of functions, defined on the whole real line, integrable on ${\mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({\mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({\mathbb R})$ by splines from $S_{h,m}\cap L_1({\mathbb R})$ (case $\alpha=\beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({\mathbb T})$ of $2\pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({\mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({\mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({\mathbb T})$.https://journals.pnu.edu.ua/index.php/cmp/article/view/5570best $l_1$-approximationone-sided approximationnon-symmetric approximationpolynomial splinefunctional class |
| spellingShingle | N.V. Parfinovych Non-symmetric approximations of functional classes by splines on the real line Karpatsʹkì Matematičnì Publìkacìï best $l_1$-approximation one-sided approximation non-symmetric approximation polynomial spline functional class |
| title | Non-symmetric approximations of functional classes by splines on the real line |
| title_full | Non-symmetric approximations of functional classes by splines on the real line |
| title_fullStr | Non-symmetric approximations of functional classes by splines on the real line |
| title_full_unstemmed | Non-symmetric approximations of functional classes by splines on the real line |
| title_short | Non-symmetric approximations of functional classes by splines on the real line |
| title_sort | non symmetric approximations of functional classes by splines on the real line |
| topic | best $l_1$-approximation one-sided approximation non-symmetric approximation polynomial spline functional class |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/5570 |
| work_keys_str_mv | AT nvparfinovych nonsymmetricapproximationsoffunctionalclassesbysplinesontherealline |