Integral characterization of elementary definitizable functions (Q810810)

Let \(S\) be a commutative semigroup with involution and identity and let \(\Gamma\) be a subset of semicharacters on S equipped with the topology of pointwise convergence. A kernel \(Log_{\rho_ 0}(s,\rho)\) such that \(Log_{\rho_ 0}(s,\rho)\) is a branch of \(\log (\rho (s)/\rho_ 0(s))\) is introduced whose imaginary part agrees with the classical Lévy- kernel for groups. The exponential kernel \(Ex_{\kappa}\) is then defined as a truncation of the power series expansion of \(\exp (Log_{\rho_ 0}(s,\rho))\). An integral representation for a class of definitizable functions with one singularity \(\rho_ 0\) is derived. When \(S\) is a group with \(s^*=s^{-1}\) or more generally an inverse semigroup and \(\rho_ 0\equiv 1\), this representation reduces to the form \[ \phi(s)=\omega(s)+\int_{\Gamma}(\rho(s)-Ex_{\kappa}(s,\rho))d\mu(\rho),\quad (s\in S,\quad \rho \in \Gamma), \] where \(\mu\) is a non-negative measure and \(\omega\) satisfies an elementary functional equation. The above formula coincides with the celebrated Lévy-Khinchin formula for conditionally positive definite functions as well as the more recent extension of Sásvári for the group case. In this special case, the \(Log_{\rho_ 0}(s,\rho)\) reduces to a branch of \(\text{arg}(\rho(s))\) and the definitizable functions introduced here are a subclass of those studied by Berg and Sásvári. For an arbitrary semigroup \(S\), the integral representation is more complicated since the kernel may have a non-zero real part. Finally, we apply our results to solve Haviland's indefinite moment problem.