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The Stability of Isometry by Singular Value Decomposition

Author

Listed:
  • 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)

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.

Suggested Citation

  • Soon-Mo Jung & Jaiok Roh, 2025. "The Stability of Isometry by Singular Value Decomposition," Mathematics, MDPI, vol. 13(15), pages 1-11, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2500-:d:1716664
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