Levi-Civita functional equations and the status of spectral synthesis on semigroups
The Levi-Civita functional equation on a semigroup \(S\) is the equation \(f(xy)=\sum_{i=1}^n g_i(x)\, h_i(y)\), \(x, y\in S\), for any positive integer \(n\) and unknown functions \(f, g_1,\dots, g_n, h_1, \dots, h_n : S\to {\mathbb C}\). One of the major results concerning the Levi-Civita equation is that on abelian groups all of its solutions are exponential polynomials. In the paper under review the authors show that on many semigroups (even commutative ones with identity) there exist solutions of the Levi-Civita equation which are not exponential polynomials. Also they identify conditions on commutative monoids insuring that all solutions of the Levi-Civita equation are exponential polynomials.
scientific article; zbMATH DE number 7395081
scientific article