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&...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-08-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/15/2500 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849405984014860288 |
|---|---|
| author | Soon-Mo Jung Jaiok Roh |
| author_facet | Soon-Mo Jung Jaiok Roh |
| author_sort | Soon-Mo Jung |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-9f07f272a3a948ffb49ccae4dbac3f93 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-9f07f272a3a948ffb49ccae4dbac3f932025-08-20T03:36:31ZengMDPI AGMathematics2227-73902025-08-011315250010.3390/math13152500The Stability of Isometry by Singular Value DecompositionSoon-Mo Jung0Jaiok Roh1Nano Convergence Technology Research Institute, School of Semiconductor & Display Technology, Hallym University, Chuncheon 24252, Republic of KoreaIlsong Liberal Art Schools (Mathematics), Hallym University, Chuncheon 24252, Republic of KoreaHyers 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.https://www.mdpi.com/2227-7390/13/15/2500stabilityisometryε-isometrybounded domainsingular value decomposition |
| spellingShingle | Soon-Mo Jung Jaiok Roh The Stability of Isometry by Singular Value Decomposition Mathematics stability isometry ε-isometry bounded domain singular value decomposition |
| title | The Stability of Isometry by Singular Value Decomposition |
| title_full | The Stability of Isometry by Singular Value Decomposition |
| title_fullStr | The Stability of Isometry by Singular Value Decomposition |
| title_full_unstemmed | The Stability of Isometry by Singular Value Decomposition |
| title_short | The Stability of Isometry by Singular Value Decomposition |
| title_sort | stability of isometry by singular value decomposition |
| topic | stability isometry ε-isometry bounded domain singular value decomposition |
| url | https://www.mdpi.com/2227-7390/13/15/2500 |
| work_keys_str_mv | AT soonmojung thestabilityofisometrybysingularvaluedecomposition AT jaiokroh thestabilityofisometrybysingularvaluedecomposition AT soonmojung stabilityofisometrybysingularvaluedecomposition AT jaiokroh stabilityofisometrybysingularvaluedecomposition |