ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER
In the space \(L(\mathbb{S}^{m-1})\) of functions integrable on the unit sphere \(\mathbb{S}^{m-1}\) of the Euclidean space \(\mathbb{R}^{m}\) of dimension \(m\ge 3\), we discuss the problem of one-sided approximation to the characteristic function of a spherical layer \(\mathbb{G}(J)=\{x=(x_1,x_2,\...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2018-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/130 |
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| author | Marina V. Deikalova Anastasiya Yu. Torgashova |
| author_facet | Marina V. Deikalova Anastasiya Yu. Torgashova |
| author_sort | Marina V. Deikalova |
| collection | DOAJ |
| description | In the space \(L(\mathbb{S}^{m-1})\) of functions integrable on the unit sphere \(\mathbb{S}^{m-1}\) of the Euclidean space \(\mathbb{R}^{m}\) of dimension \(m\ge 3\), we discuss the problem of one-sided approximation to the characteristic function of a spherical layer \(\mathbb{G}(J)=\{x=(x_1,x_2,\ldots,x_m)\in \mathbb{S}^{m-1}\colon x_m\in J\},\) where \(J\) is one of the intervals \((a,1],\) \((a,b),\) and \([-1,b),\) \(-1< a<b< 1,\) by the set of algebraic polynomials of given degree \(n\) in \(m\) variables. This problem reduces to the one-dimensional problem of one-sided approximation in the space \(L^\phi(-1,1)\) with the ultraspherical weight \(\phi(t)=(1-t^2)^\alpha,\ \alpha=(m-3)/2,\) to the characteristic function of the interval \(J\). This result gives a solution of the problem of one-sided approximation to the characteristic function of a spherical layer in all cases when a solution of the corresponding one-dimensional problem known. In the present paper, we use results by A.G.Babenko, M.V.Deikalova, and Sz.G.Revesz (2015) and M.V.Deikalova and A.Yu.Torgashova (2018) on the one-sided approximation to the characteristic functions of intervals. |
| format | Article |
| id | doaj-art-d24a004f8abb43edab181e32f4e4c9b6 |
| institution | Matheson Library |
| issn | 2414-3952 |
| language | English |
| publishDate | 2018-12-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-d24a004f8abb43edab181e32f4e4c9b62025-08-02T20:08:50ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522018-12-014210.15826/umj.2018.2.00362ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYERMarina V. Deikalova0Anastasiya Yu. Torgashova1Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000In the space \(L(\mathbb{S}^{m-1})\) of functions integrable on the unit sphere \(\mathbb{S}^{m-1}\) of the Euclidean space \(\mathbb{R}^{m}\) of dimension \(m\ge 3\), we discuss the problem of one-sided approximation to the characteristic function of a spherical layer \(\mathbb{G}(J)=\{x=(x_1,x_2,\ldots,x_m)\in \mathbb{S}^{m-1}\colon x_m\in J\},\) where \(J\) is one of the intervals \((a,1],\) \((a,b),\) and \([-1,b),\) \(-1< a<b< 1,\) by the set of algebraic polynomials of given degree \(n\) in \(m\) variables. This problem reduces to the one-dimensional problem of one-sided approximation in the space \(L^\phi(-1,1)\) with the ultraspherical weight \(\phi(t)=(1-t^2)^\alpha,\ \alpha=(m-3)/2,\) to the characteristic function of the interval \(J\). This result gives a solution of the problem of one-sided approximation to the characteristic function of a spherical layer in all cases when a solution of the corresponding one-dimensional problem known. In the present paper, we use results by A.G.Babenko, M.V.Deikalova, and Sz.G.Revesz (2015) and M.V.Deikalova and A.Yu.Torgashova (2018) on the one-sided approximation to the characteristic functions of intervals.https://umjuran.ru/index.php/umj/article/view/130One-sided approximationCharacteristic functionSpherical layerSpherical capAlgebraic polynomials |
| spellingShingle | Marina V. Deikalova Anastasiya Yu. Torgashova ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER Ural Mathematical Journal One-sided approximation Characteristic function Spherical layer Spherical cap Algebraic polynomials |
| title | ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER |
| title_full | ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER |
| title_fullStr | ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER |
| title_full_unstemmed | ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER |
| title_short | ONE-SIDED \(L\)-APPROXIMATION ON A SPHERE OF THE CHARACTERISTIC FUNCTION OF A LAYER |
| title_sort | one sided l approximation on a sphere of the characteristic function of a layer |
| topic | One-sided approximation Characteristic function Spherical layer Spherical cap Algebraic polynomials |
| url | https://umjuran.ru/index.php/umj/article/view/130 |
| work_keys_str_mv | AT marinavdeikalova onesidedlapproximationonasphereofthecharacteristicfunctionofalayer AT anastasiyayutorgashova onesidedlapproximationonasphereofthecharacteristicfunctionofalayer |