A Summary Equation for Functions Holomorphic Outside Two Quadrangles, with Application

Abstract

We consider a four-element summary equation in the class of functions holomorphic outside two quadrangles. The solution is an odd function having a zero of multiplicity at least three at infinity. 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 over the boundary of the quadrangles. We suggest a method for regularizing the summary equation. The method substantially relies on a Carleman involutive shift that maps each side into itself and changes its orientation. Moreover, the midpoints of the sides are fixed points of the said shift. Furthermore, we establish a condition for the equivalence of the regularized equation. By using the contraction mapping theorem in a Banach space, we single out some special cases in which the obtained Fredholm equation of the second kind is solvable. We also indicate several applications to interpolation problems for entire functions of exponential type. Those problems can be seen as a generalization of the Stieltjes moment problem to the case of two rays, wherein a piecewise exponential weight function arises.