an:03966950
Zbl 0599.39007
Skof, Fulvia
Local properties and approximation of operators
IT
Rend. Sem. Mat. Fis. Milano 53, 113-129 (1983).
00152448
1983
j
39B52
Hyers stability; local uniform approximation; Banach space
This paper is connected with the theory of functional equations in the meaning of J. Acz??l and more exactly with their Hyers stability [cf. \textit{D. H. Hyers}, Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)]. We state some results and problems concerning the local uniform approximation and the extension of an operator \(f: D_ f\subset {\mathbb{R}}\to X\) (X being a Banach space) for which the condition \(\| f(x+y)-f(x)-f(y)\| <\delta\) holds only in a given subset of \({\mathbb{R}}^ 2\) for some \(\delta >0\). Similar problems are posed in relation to the condition \(\| f(x+y)+f(x-y)-2f(x)-2f(y)\| <\delta\).
Zbl 0061.264