Superstability of the generalized orthogonality equation on restricted domains |
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Authors: | Soon-Mo Jung Prasanna K Sahoo |
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Institution: | (1) Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, Korea;(2) Department of Mathematics, University of Louisville, 40292 Louisville, KY, USA |
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Abstract: | Chmielinski has proved in the paper 4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD
n be a suitable subset of ℝn. If a function f:D
n → ℝn satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D
n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D
n. |
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Keywords: | Superstability generalized orthogonality equation |
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