On Cauchy-type functional equations (Q1774706)

The functional equations \[ \int_{G} f(xty)d \mu (t)= \sum_{i=1}^{n} g_{i}(x)h_{i}(y), \; x,y \in G, \; n \in \mathbb N \tag{1} \] are studied over a Hausdorff topological locally compact group \(G\). \(\mu\)-spherical and \(\mu\)-invariant functions are used to obtain the main result (\(\mu\) is a complex bounded measure on \(G\)). Under the assumptions that \(G\) is compact and \(\mu\) is Gelfand measure the authors give necessary and sufficient conditions for (1) to have solutions. \smallskip Complete lists of functions satisfying the functional equations \[ \int_{G} f(xty)d \mu (t)= f(x)g(y)+f(y)g(x), \; x,y \in G, \] \[ \int_{G} f(xty)d \mu (t)= f(x)f(y)+g(y)g(x), \; x,y \in G \] are also given.