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 $\mathbb{Q}_p$.
Real Anal. Exchange
31(1):
125-132
(2005-2006).
C. J. Everett and S. M. Ulam, On Some Possibilities of Generalizing the Lorentz Group in the Special Relativity Theory, J. of Comb. Theory, 1 (1966), 248–270. MR198925 10.1016/S0021-9800(66)80031-5 C. J. Everett and S. M. Ulam, On Some Possibilities of Generalizing the Lorentz Group in the Special Relativity Theory, J. of Comb. Theory, 1 (1966), 248–270. MR198925 10.1016/S0021-9800(66)80031-5
G. Forti, Hyers-Ulam Stability of Functional Equations in Several Variables, Aequationes Mathematicae, 50 (1995), 143–190. MR1336866 0836.39007 10.1007/BF01831117 G. Forti, Hyers-Ulam Stability of Functional Equations in Several Variables, Aequationes Mathematicae, 50 (1995), 143–190. MR1336866 0836.39007 10.1007/BF01831117
D. H. Hyers, On the stability of the linear functional equation, Proc. N.A.S., 27 (1941), 222–224. MR4076 10.1073/pnas.27.4.222 D. H. Hyers, On the stability of the linear functional equation, Proc. N.A.S., 27 (1941), 222–224. MR4076 10.1073/pnas.27.4.222
D. H. Hyers and T. Rassias, Approximate Homomorphisms, Aequationes Mathematicae, 44 (1992), 125–153. MR1181264 0806.47056 10.1007/BF01830975 D. H. Hyers and T. Rassias, Approximate Homomorphisms, Aequationes Mathematicae, 44 (1992), 125–153. MR1181264 0806.47056 10.1007/BF01830975
A. Khrennikov, $p$-adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers, (1994). MR1325924 0833.46061 A. Khrennikov, $p$-adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers, (1994). MR1325924 0833.46061
A. Khrennikov, Non–Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, (1997). MR1746953 0920.11087 A. Khrennikov, Non–Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, (1997). MR1746953 0920.11087
L. Székelyhidi, Ulam's Problem, Hyers Solution-and to Where they led, Stability of Functional Equations, Kluwer Academic Publishers, (2000), 259–285. MR1792088 L. Székelyhidi, Ulam's Problem, Hyers Solution-and to Where they led, Stability of Functional Equations, Kluwer Academic Publishers, (2000), 259–285. MR1792088
S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, Inc., (1960) (2000), 63–69. MR120127 0086.24101S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, Inc., (1960) (2000), 63–69. MR120127 0086.24101