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Local stability of the functional equation characterizing polynomial functions. (English) Zbl 0717.39007
Let \((X,\|\|)\) be a Banach space, and p a nonnegative integer. A function g: \({\mathbb{R}}\to X\) is said to be a polynomial of p-th order if it satisfies the equation \(\Delta_ h^{p+1}g(x)=0\) for all x and h in \({\mathbb{R}}\). The equation is known to be stable in the sense that if f: \({\mathbb{R}}\to X\) satisfies the inequality (1) \(\| \Delta_ h^{p+1}f(x)\| \leq \epsilon\) for all x and h in \({\mathbb{R}}\), then f is uniformly close to a polynomial of p-th order. For local stability, the inequality is assumed to hold only on a restricted domain.
The author proves several such results, the following being typical. For each p, there is a constant \(\ell_ p\) with the following property. Let \(\epsilon\) and a be positive numbers, \(x_ 0\) real, and let I denote the interval \((x_ 0-a\), \(x_ 0+a)\). Let f: \(I\to X\) satisfy the inequality (1) for those x in I and h in (-a,a) with \(x+(p+1)h\) in I. Then there exists a polynomial g: \({\mathbb{R}}\to X\) of p-th order such that \(\| f(x)-g(x)\| \leq \ell_ p\epsilon\) for x in I.
In the course of his considerations, the author obtains theorems concerning approximately multiadditive functions on restricted domains.
Reviewer: F.-W.Carroll

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