The Stability of Isometry by Singular Value Decomposition
Hyers and Ulam considered the problem of whether there is a true isometry that approximates the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula&...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-08-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/15/2500 |
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| Summary: | Hyers and Ulam considered the problem of whether there is a true isometry that approximates the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-isometry defined on a Hilbert space with a stability constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>10</mn><mi>ε</mi></mrow></semantics></math></inline-formula>. Subsequently, Fickett considered the same question on a bounded subset of the <i>n</i>-dimensional Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> with a stability constant of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>27</mn><msup><mi>ε</mi><mrow><mn>1</mn><mo>/</mo><msup><mn>2</mn><mi>n</mi></msup></mrow></msup></mrow></semantics></math></inline-formula>. And Vestfrid gave a stability constant of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>27</mn><mi>n</mi><mi>ε</mi></mrow></semantics></math></inline-formula> as the answer for bounded subsets. In this paper, by applying singular value decomposition, we improve the previous stability constants by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msqrt><mi>n</mi></msqrt><mi>ε</mi></mrow></semantics></math></inline-formula> for bounded subsets, where the constant <i>C</i> depends on the approximate linearity parameter <i>K</i>, which is defined later. |
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| ISSN: | 2227-7390 |