Real Analysis Exchange

Stability of the Cauchy functional equation over p-adic fields.

L. M. Arriola and W. A. Beyer

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This paper corrects some errors found in [2], which discusses an extension of Lorentz transformations over a non-Archimedean valued field; namely, the $p$-adic field $\mathbb{Q}_{p}$. The paper [2] is based on the results given by Hyers [7] which showed that for a continuous function $f$ defined on $\mathbb{R}$, the Cauchy functional equation $f(x + y) = f(x) + f(y)$ is stable. By stable we mean that if there exists $\epsilon >0$ such that $\| f(x + y) - f(x) - f(y) \| < \epsilon$, $\forall x,y$, then there exists a unique and continuous $\mathcal{L}$ such that $\| \mathcal{L}(x) - f(x) \| \leq \epsilon$, $\forall x$ and $\mathcal{L}(x + y) = \mathcal{L}(x) + \mathcal{L}(y)$. In this paper, we show this result is true on the $p$-adic field $\mathbb{Q}_p$.

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 125-132.

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 39B22: Equations for real functions [See also 26A51, 26B25]

Cauchy functional equation p-adic field stability


Arriola, L. M.; Beyer, W. A. Stability of the Cauchy functional equation over p -adic fields. Real Anal. Exchange 31 (2005), no. 1, 125--132.

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