Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases
This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space <i>X</i> with a symmetric basis. We obtain some positive results for the case when <i>X</i> admits a separating polynomial using a symmetrizati...
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| Format: | Article |
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MDPI AG
2021-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/13/12/2318 |
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| author | Mariia Martsinkiv Andriy Zagorodnyuk |
| author_facet | Mariia Martsinkiv Andriy Zagorodnyuk |
| author_sort | Mariia Martsinkiv |
| collection | DOAJ |
| description | This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space <i>X</i> with a symmetric basis. We obtain some positive results for the case when <i>X</i> admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, we introduce <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-slice <i>G</i>-analytic functions that have a behavior similar to <i>G</i>-analytic functions at points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> such that all coordinates of <i>x</i> are greater than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi></mrow></semantics></math></inline-formula>, and we prove a theorem on approximations of uniformly continuous functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>0</mn></msub></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-slice <i>G</i>-analytic functions. |
| first_indexed | 2024-03-10T03:01:01Z |
| format | Article |
| id | doaj.art-a31cbc26355a486bac634631ce42948d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T03:01:01Z |
| publishDate | 2021-12-01 |
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| series | Symmetry |
| spelling | doaj.art-a31cbc26355a486bac634631ce42948d2023-11-23T10:45:24ZengMDPI AGSymmetry2073-89942021-12-011312231810.3390/sym13122318Approximations of Symmetric Functions on Banach Spaces with Symmetric BasesMariia Martsinkiv0Andriy Zagorodnyuk1Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, UkraineFaculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, UkraineThis paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space <i>X</i> with a symmetric basis. We obtain some positive results for the case when <i>X</i> admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow></semantics></math></inline-formula>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>, we introduce <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-slice <i>G</i>-analytic functions that have a behavior similar to <i>G</i>-analytic functions at points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> such that all coordinates of <i>x</i> are greater than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi></mrow></semantics></math></inline-formula>, and we prove a theorem on approximations of uniformly continuous functions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>0</mn></msub></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-slice <i>G</i>-analytic functions.https://www.mdpi.com/2073-8994/13/12/2318symmetric functions on Banach spacesapproximations by analytic functionsinvariant means |
| spellingShingle | Mariia Martsinkiv Andriy Zagorodnyuk Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases Symmetry symmetric functions on Banach spaces approximations by analytic functions invariant means |
| title | Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases |
| title_full | Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases |
| title_fullStr | Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases |
| title_full_unstemmed | Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases |
| title_short | Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases |
| title_sort | approximations of symmetric functions on banach spaces with symmetric bases |
| topic | symmetric functions on Banach spaces approximations by analytic functions invariant means |
| url | https://www.mdpi.com/2073-8994/13/12/2318 |
| work_keys_str_mv | AT mariiamartsinkiv approximationsofsymmetricfunctionsonbanachspaceswithsymmetricbases AT andriyzagorodnyuk approximationsofsymmetricfunctionsonbanachspaceswithsymmetricbases |