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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

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