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.

中文翻译：

---

两个四边形外的全纯函数的汇总方程及其应用

摘要

我们在两个四边形之外的全纯函数类中考虑一个四元素汇总方程。该解是一个奇数函数，在零处具有至少三个的多重性零。边界值在任何不包含顶点的压缩体上都满足Hölder条件。在顶点处，我们最多允许对数奇点。我们寻找四边形边界上的柯西型积分形式的解决方案。我们建议一种用于对汇总方程进行正则化的方法。该方法基本上依赖于Carleman渐进式移位，该移位将每侧映射到自身并改变其方向。此外，侧面的中点是所述偏移的固定点。此外，我们为正则方程的等价性建立了条件。通过在Banach空间中使用收缩映射定理，我们挑出一些特殊情况，其中所获得的第二类Fredholm方程是可解的。我们还指出了指数型整个函数对插值问题的几种应用。这些问题可以看作是Stieltjes矩问题到两束射线的情况的推广，其中出现了分段指数权函数。