Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings
This article primarily examines the approximation properties of a weighted Sobolev space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mrow><mn>2</mn><mo>...
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2025-01-01
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| author | Jiayi Qiu Guanggui Chen Yanyan Xu Ying Luo Hang Ren |
| author_facet | Jiayi Qiu Guanggui Chen Yanyan Xu Ying Luo Hang Ren |
| author_sort | Jiayi Qiu |
| collection | DOAJ |
| description | This article primarily examines the approximation properties of a weighted Sobolev space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mrow><mn>2</mn><mo>,</mo><mi>κ</mi></mrow><mi>r</mi></msubsup></semantics></math></inline-formula> defined on a sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> equipped with Gaussian measures. Specifically, this study focuses on both the average case and probabilistic case settings. The exact asymptotic orders of the Gel’fand <i>n</i>-width and the linear <i>n</i>-width of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mrow><mn>2</mn><mo>,</mo><mi>κ</mi></mrow><mi>r</mi></msubsup></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> are derived for these settings, providing a comprehensive understanding of their approximation characteristics. |
| format | Article |
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| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-af58e850e7994590943932f27ce5bbfd2025-01-24T13:22:14ZengMDPI AGAxioms2075-16802025-01-011414210.3390/axioms14010042Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different SettingsJiayi Qiu0Guanggui Chen1Yanyan Xu2Ying Luo3Hang Ren4School of Science, Xihua University, Chengdu 610039, ChinaYibin Campus, Xihua University, Yibin 644005, ChinaSchool of Science, Xihua University, Chengdu 610039, ChinaSchool of Science, Xihua University, Chengdu 610039, ChinaSchool of Science, Xihua University, Chengdu 610039, ChinaThis article primarily examines the approximation properties of a weighted Sobolev space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mrow><mn>2</mn><mo>,</mo><mi>κ</mi></mrow><mi>r</mi></msubsup></semantics></math></inline-formula> defined on a sphere <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> equipped with Gaussian measures. Specifically, this study focuses on both the average case and probabilistic case settings. The exact asymptotic orders of the Gel’fand <i>n</i>-width and the linear <i>n</i>-width of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mrow><mn>2</mn><mo>,</mo><mi>κ</mi></mrow><mi>r</mi></msubsup></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">S</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> are derived for these settings, providing a comprehensive understanding of their approximation characteristics.https://www.mdpi.com/2075-1680/14/1/42Gel’fand widthweighted Sobolev classessphereGaussian measure |
| spellingShingle | Jiayi Qiu Guanggui Chen Yanyan Xu Ying Luo Hang Ren Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings Axioms Gel’fand width weighted Sobolev classes sphere Gaussian measure |
| title | Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings |
| title_full | Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings |
| title_fullStr | Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings |
| title_full_unstemmed | Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings |
| title_short | Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings |
| title_sort | approximation characteristics of weighted sobolev spaces on sphere in different settings |
| topic | Gel’fand width weighted Sobolev classes sphere Gaussian measure |
| url | https://www.mdpi.com/2075-1680/14/1/42 |
| work_keys_str_mv | AT jiayiqiu approximationcharacteristicsofweightedsobolevspacesonsphereindifferentsettings AT guangguichen approximationcharacteristicsofweightedsobolevspacesonsphereindifferentsettings AT yanyanxu approximationcharacteristicsofweightedsobolevspacesonsphereindifferentsettings AT yingluo approximationcharacteristicsofweightedsobolevspacesonsphereindifferentsettings AT hangren approximationcharacteristicsofweightedsobolevspacesonsphereindifferentsettings |