Almost periodicity and functional equations
From author's introduction: ''In this note we present some applications of Fourier analysis in the theory of functional equations. The basic idea is that we try to prove the almost periodicity of the solution instead of differentiability, and the basic observation is that from this point of view ''boundedness'' is a good condition for the solution, playing the role of ''measurability''. In other words, it turns out that some types of functional equations on topological groups have the property that all bounded continuous solutions are almost periodic, and by Fourier transformation the solutions can be determined effectively from simple linear systems of equations.'' The author proves the following theorem. Let G be a topological group, \(F: G\times G\to {\mathbb{C}}\) be an almost periodic function, \(f,g_ i,k_ i: G\to K\) \((i=1,2,...,n)\) be mappings such that \((1)\quad f(xy)=\sum_{i}g_ i(x)h_ i(y)+F(x,y)\) holds. If each \(g_ i\) is bounded and continuous and each \(h_ i\) is bounded, then f is almost periodic. The author applies this theorem to obtain continuous bounded solutions of \(f(xy)=f(x)g(y)+f(y)g(x),\quad f(xy)=f(x)f(y)-g(x)g(y),\) and \(f(xy)+f(xy^{-1})=2f(x)f(y)\) on a topological Abelian group G with a separating family \(G^*\).
scientific article; zbMATH DE number 3903218