Sum-difference equation for analytic functions generated by triangle and its applications

Abstract

Let $ D $ be a triangle and $ \Gamma $ by the half of its boundary $ \partial D $. We consider an element-wise linear sum-difference equation in the class of functions holomorphic outside $ \Gamma $ and vanishing at infinity. The solution is sought in the form of a Cauchy-type integral over $ \Gamma $ with an unknown density. The boundary values satisfy the Hölder condition on each compact subset in $ \Gamma $ containing no nodes. At most logarithmic singularities are admitted at the nodes. In order to regularize the equation to $ \partial D $, we introduce a piecewise linear Carleman shift. It maps each side into itself changing the orientation. In this case, the midpoints of the sides are fixed points. We regularize the equation and find its solvability condition for. We consider a particular case when the number of solvability conditions can be counted exactly. We provide applications to interpolation problems for entire functions of exponential type. Previously, similar problems were investigated for tetragon, pentagon, and hexagon.