Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

A general functional equation and its stability

Author(s): John A. Baker
Journal: Proc. Amer. Math. Soc. 133 (2005), 1657-1664.
MSC (2000): Primary 39B72, 39B52; Secondary 39B05
Posted: January 13, 2005
MathSciNet review: 2120259
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Abstract | References | Similar articles | Additional information

Abstract: Suppose that and are vector spaces over $\mathbb{Q}, \mathbb{R}$ or $\mathbb{C}$ and $\alpha_0,\beta_0,\dots,\alpha_m,\beta_m$ are scalar such that $\alpha_j\beta_k-\alpha_k\beta_j\neq 0$ whenever $0\leq j<k\leq m.$ We prove that if $f_k:V\rightarrow B$ for $0\leq k\leq m$ and

$\begin{displaymath}(*)\quad\qquad\qquad\qquad \sum^m_{k=0} f_k(\alpha_kx+\beta_ky)=0\quad \text{for all} x,y\in V, \qquad\quad\qquad\qquad\quad \end{displaymath}$

then each

is a ``generalized'' polynomial map of ``degree'' at most

In case $V=\mathbb{R}^n$ and $B=\mathbb{C}$ we show that if some is bounded on a set of positive inner Lebesgue measure, then it is a genuine polynomial function.

Our main aim is to establish the stability of (in the sense of Ulam) in case is a Banach space.

We also solve a distributional analogue of and prove a mean value theorem concerning harmonic functions in two real variables.

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