A class of linear functionals on the Bergman space over the complement of the unit lattice (Q2642853)

Let \(\Gamma\) denote the integer lattice on the complex plane \(\mathbb C\), consisting of all points of the form \(n+mi\), where \(n\) and \(m\) are arbitrary integers. Let \(A(\Gamma)\) denote the Bergman subspace of \(L^1(\mathbb C,dA)\) consisting of functions that are analytic in \(\mathbb C-\Gamma\), where \(dA\) is Lebesgue area measure. The main result of the paper states that if \(f\) is a rational function, not identically zero, then the linear functional \[ \Lambda(g)=\int_{\mathbb C} g\,{| f| \over f}\,dA,\quad g\in A(\Gamma), \] has norm \(1\). Several corollaries are obtained.