The Stability of Isometry by Singular Value Decomposition

Soon-Mo Jung and Jaiok Roh ()
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Soon-Mo Jung: Nano Convergence Technology Research Institute, School of Semiconductor & Display Technology, Hallym University, Chuncheon 24252, Republic of Korea
Jaiok Roh: Ilsong Liberal Art Schools (Mathematics), Hallym University, Chuncheon 24252, Republic of Korea

Mathematics, 2025, vol. 13, issue 15, 1-11

Abstract: Hyers and Ulam considered the problem of whether there is a true isometry that approximates the ε -isometry defined on a Hilbert space with a stability constant 10 ε . Subsequently, Fickett considered the same question on a bounded subset of the n -dimensional Euclidean space R n with a stability constant of 27 ε 1 / 2 n . And Vestfrid gave a stability constant of 27 n ε as the answer for bounded subsets. In this paper, by applying singular value decomposition, we improve the previous stability constants by C n ε for bounded subsets, where the constant C depends on the approximate linearity parameter K , which is defined later.

Keywords: stability; isometry; ε-isometry; bounded domain; singular value decomposition (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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