On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings on the Gaussian plane
Let \(Z\), \(W\) be complex linear spaces and \(Q:Z\to W\). The functional equation \[ Q\Biggl(\sum_{i=1}^na_iz_i\Biggr)+\sum_{1\leq i<j\leq n} (a_j\bar{z}_i-a_i\bar{z}_j) =m\sum_{i=1}^nQ(z_i) \] for every complex \(z_i\in Z\) \((i=1,2,\dots,n)\) and fixed \({\mathbb C}^n\ni a=(a_1,a_2,\dots,a_n) \neq (0,0,\dots,0)\) with \[ 0<m=\sum_{i=1}^n|a_i|^2\neq\frac{1}{n}\Biggl[1+\binom{n}{2}\Biggr] \] is introduced. Its solutions are called \textit{quadratic mappings with respect to~\(a\)}. The Hyers-Ulam stability problem for such mappings is solved.
scientific article; zbMATH DE number 1947334