On moment problem for entire functions generated by doubly periodic group

Abstract

We consider a lacunar problem for Stieltjes moments with an exponential weight. The solution is sought in the class of entire functions of exponential type, the indicator diagram of which is a some square. We construct nontrivial solutions of the corresponding homogeneous problem. This problem is reduced to the study of a linear total equation in the class of functions holomorphic outside four squares. At infinity, they have zero of a multiplicity at least three. Their boundary values satisfy the Hölder condition on any compact set containing no square vertices. At most logarithmic singularities are allowed at these vertices. The solution is sought in the form of an Cauchy type integral over the boundary of these squares with an unknown density. A method for regularizing the total equation is proposed. The condition of equivalence of this regularization is clarified. We find particular case when the obtained Fredholm equation of the second kind is solvable. In order to do this, we employ the principle of contracting mappings in a Banach space.