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On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings on the Gaussian plane. (English) Zbl 1025.39018

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 quadratic mappings with respect to \(a\). The Hyers-Ulam stability problem for such mappings is solved.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B32 Functional equations for complex functions
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