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A general functional equation and its stability

Author: John A. Baker
Journal: Proc. Amer. Math. Soc. 133 (2005), 1657-1664
MSC (2000): Primary 39B72, 39B52; Secondary 39B05
Published electronically: January 13, 2005
MathSciNet review: 2120259
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Abstract: Suppose that $V$ and $B$ 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 $f_k$ is a ``generalized'' polynomial map of ``degree'' at most $m-1.$

In case $V=\mathbb{R}^n$ and $B=\mathbb{C}$ we show that if some $f_k$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 $B$ 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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Additional Information

John A. Baker
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Keywords: Functional equation, stability
Received by editor(s): April 25, 2003
Published electronically: January 13, 2005
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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