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Cauchy’s functional equation, Schur’s lemma, one-dimensional special relativity, and Möbius’s functional equation. (English) Zbl 1401.39027

Daras, Nicholas J. (ed.) et al., Modern discrete mathematics and analysis. With applications in cryptography, information systems and modeling. Cham: Springer (ISBN 978-3-319-74324-0/hbk; 978-3-319-74325-7/ebook). Springer Optimization and Its Applications 131, 389-396 (2018).
Summary: This article explores a remarkable connection between Cauchy’s functional equation, Schur’s lemma in representation theory, the one-dimensional relativistic velocities in special relativity, and Möbius’s functional equation. Möbius’s exponential equation is a functional equation defined by \[ f(a\oplus_M b) = f(a)f(b), \] where \(\oplus_M\) is Möbius addition given by \(a\oplus_M b = \dfrac {a+b}{1+\bar {a}b}\) for all complex numbers \(a\) and \(b\) of modulus less than one, and the product \(f(a)f(b)\) is taken in the field of complex numbers. We indicate that, in some sense, Möbius’s exponential equation is an extension of Cauchy’s exponential equation. We also exhibit a one-to-one correspondence between the irreducible linear representations of an abelian group on a complex vector space and the solutions of Cauchy’s exponential equation and extend this to the case of Möbius’s exponential equation. We then give the complete family of Borel measurable solutions to Cauchy’s exponential equation with domain as the group of one-dimensional relativistic velocities under the restriction of Möbius addition.
For the entire collection see [Zbl 1403.05004].

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
20K01 Finite abelian groups
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