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On a modified Hyers-Ulam sequence. (English) Zbl 0746.46038

It is shown that, given a mapping \(f: E\to F\) between Banach spaces, which is continuous along rays through 0, and for which there is a \(\delta\geq 0\) and a \(0\leq p<1\) such that \[ \| f(x+y)-f(x)- f(y)\|\leq\delta\cdot(\| x\|^ p+\| y\|^ p)\text{ for all } x,y, \] there is a unique linear mapping \(T: E\to F\) such that for all integers \(k\geq 2\) one has \[ \| f(x)-T(x)\|\leq\delta\cdot\| x\|^ p\cdot {{k+\sum_{j=2}^{k-1}j^ k} \over {k-k^ p}} \text{ for all } x. \] In fact \(T(x)=\lim_{n\to\infty} {{f(k^ n x)}\over{k^ n}}\). The author poses the question whether the minimum is attained for \(k=2\).
Reviewer: A.Kriegl (Wien)

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
41A35 Approximation by operators (in particular, by integral operators)
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[1] Hyers, D. H., On the stability of the linear functional equation, (Proc. Natl. Acad. Sci. U.S.A., 27 (1941)), 222-224 · Zbl 0061.26403
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