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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Main Authors: Jiayi Qiu, Guanggui Chen, Yanyan Xu, Ying Luo, Hang Ren
Format: Article
Language:English
Published: MDPI AG 2025-01-01
Series:Axioms
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Online Access:https://www.mdpi.com/2075-1680/14/1/42
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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.
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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