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Local properties and approximation of operators. (Italian. English summary) Zbl 0599.39007

This paper is connected with the theory of functional equations in the meaning of J. Aczél and more exactly with their Hyers stability [cf. 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 \(\| f(x+y)+f(x-y)-2f(x)-2f(y)\| <\delta\).

MSC:

39B52 Functional equations for functions with more general domains and/or ranges

Citations:

Zbl 0061.264
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References:

[1] Fenyö, I., Osservazioni su alcuni teoremi di D. H. Hyers, Istit. Lombardo Accad. Sci. Lett. Rend., A, 114, 235-242 (1980) · Zbl 0513.39008
[2] Hyers, D. H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27, 222-224 (1941) · JFM 67.0424.01
[3] Skof F.,Sull’approssimazione delle applicazioni localmente δ-additive. Atti Accad. Sci. Torino,117 (1983) (in corso di stampa).
[4] Skof F.,Sull’approssimazione delle funzioni δ-quadratiche su un dominio ristretto. (In preparazione).
[5] Tabacco A.,Equazioni funzionali su dominio ristretto. Dissertazione, Univ. Torino (1982).
[6] Trombetta B.,Proprietà di stabilità per le equazioni funzionali. Dissertazione, Univ. Torino (1983).
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