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: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/42 |
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| Summary: | 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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| ISSN: | 2075-1680 |