On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums
In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup&g...
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MDPI AG
2023-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/8/763 |
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| author | Dmytro Bushev Inna Kal’chuk |
| author_facet | Dmytro Bushev Inna Kal’chuk |
| author_sort | Dmytro Bushev |
| collection | DOAJ |
| description | In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> 2<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact upper bounds of the best approximations of this class by trigonometric polynomials. In addition, we obtain the sets of all of extremal functions for approximations of the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> by linear methods of summation of Fourier series. Furthermore, we receive the set of all extremal functions for the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> in the Korneichuk–Stechkin lemma and its analogue, the Stepanets lemma, for the Hölder class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> functions of two variables being 2<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>-periodic in each variable. |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T00:07:22Z |
| publishDate | 2023-08-01 |
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| series | Axioms |
| spelling | doaj.art-527411da225e40fe81bae17fa240ef232023-11-19T00:14:53ZengMDPI AGAxioms2075-16802023-08-0112876310.3390/axioms12080763On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard SumsDmytro Bushev0Inna Kal’chuk1Faculty of Information Technologies and Mathematics, Lesya Ukrainka Volyn National University, 43025 Lutsk, UkraineFaculty of Information Technologies and Mathematics, Lesya Ukrainka Volyn National University, 43025 Lutsk, UkraineIn this paper, we find the sets of all extremal functions for approximations of the Hölder classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> 2<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact upper bounds of the best approximations of this class by trigonometric polynomials. In addition, we obtain the sets of all of extremal functions for approximations of the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> by linear methods of summation of Fourier series. Furthermore, we receive the set of all extremal functions for the class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mn>1</mn></msup></semantics></math></inline-formula> in the Korneichuk–Stechkin lemma and its analogue, the Stepanets lemma, for the Hölder class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> functions of two variables being 2<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>-periodic in each variable.https://www.mdpi.com/2075-1680/12/8/763Favard sumsbest approximationexact upper boundsextremal functionsuniform metric |
| spellingShingle | Dmytro Bushev Inna Kal’chuk On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums Axioms Favard sums best approximation exact upper bounds extremal functions uniform metric |
| title | On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums |
| title_full | On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums |
| title_fullStr | On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums |
| title_full_unstemmed | On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums |
| title_short | On the Realization of Exact Upper Bounds of the Best Approximations on the Classes <i>H</i><sup>1,1</sup> by Favard Sums |
| title_sort | on the realization of exact upper bounds of the best approximations on the classes i h i sup 1 1 sup by favard sums |
| topic | Favard sums best approximation exact upper bounds extremal functions uniform metric |
| url | https://www.mdpi.com/2075-1680/12/8/763 |
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