Superstability of the generalized orthogonality equation on restricted domains

Abstract

Chmieliński 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 Chmieliński by proving a theorem: Let $D_n$ be a suitable subset of ${\mathbb{R}}^n$. If a function $f:D_n\to {\mathbb{R}}^n$ satisfies the inequality $||<f(x),f(y)>|-|<x,y>||\le \varphi (x,y)$ for an appropriate control function $\varphi (x,y)$ and for all $x,y\in D_n$, then $f$ satisfies the generalized orthogonality equation for any $x,y\in D_n$.

arXiv comment15 pages