The Cauchy functional equation as an initial value problem, homomorphisms, and tori

Details The Cauchy functional equation as an initial value problem, homomorphisms, and tori The Cauchy functional equation as an initial value problem, homomorphisms, and tori

2010-02-19

arxiv.org/abs/1002.3721v2

Mathematics

Classical Analysis and ODEs

V1: 4 pages. V2: 6 pages, the title was modified, a theorem about homomorphisms and tori was added, many references were added

Scientific paper

The Cauchy functional equation f(x+y)=f(x)+f(y) has been investigated by many authors, under various "regularity" conditions. We present a new method for solving this equation assuming only that a complex exponent of the unknown function is locally measurable. A key idea is to consider the equation as an initial value problem. The (rather simple) proof can be generalized, e.g., to a more abstract setting. As a by-product of this approach we prove the following theorem: given an additive homomorphism from a finite dimensional flat torus to the real line, if a complex exponent of it is measurable, then the homomorphism must vanish identically.

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