The summary equation for functions analytical outside four squares. Applications

We consider the lacunary Stieltjes moment problem
$$ \int\limits_{0}^{\infty} F(x) x^{4n+1} \exp(-x) dx=\beta_n, \ n=0,1,2. $$
We search for a solution in the class of entire functions of the exponential type that satisfy the condition $F(iz)=F(z)$. Their indicator diagram is a certain octagon. We construct nontrivial solutions to the corresponding homogeneous problem. The problem reduces studying a linear summary equation in the class of functions holomorphic outside four squares. At infinity, they have a zero of multiplicity at least three. The boundary values satisfy a Hölder condition on any compact that does not contain the vertices. At the vertices, we allow at most logarithmic singularities. We search for a solution in the form of a Cauchy-type integral with an unknown density over the boundary of those squares. We suggest a method for the regularization of the summary equation. An equivalence condition for this regularization is established. Additionally, we identify some special cases, in which the obtained Fredholm equation of the second kind is solvable. For this, we use the contraction mapping theorem in a Banach space.