Sum-Difference Equation for Analytic Functions, Generated by a Hexagon, and Its Applications

Abstract

Let D be a hexagon having a pair of parallel sides, equal in length, and L be a half of its boundary. We study solvability of a seven-element sum-difference equation equation in the class of functions holomorphic outside L and vanishing at infinity. Their boundary values satisfy the Hölder condition on every compact not containing the nodes. At the nodes they have at most logarithmic singularities. To regularize the equation, on the boundary of the hexagon we introduce a Carleman shift having jump discontinuity at the vertices. We seek a solution in the form of Cauchy type integral over L with an unknown density. We find conditions providing equivalence of such regularization. We also consider a particular case for which the corresponding Fredholm equation is solvable. We give some applications to the moment problem for entire functions of exponential type (EFETs). In particular, we construct a system of EFETs biorthogonal, with a piece-wise quasipolynomial weight, to a system of three degree functions on three rays. For such EFETs, the conjugate diagram is an octagon. We note that various generalizations of our investigations are possible due to the large arbitrariness in the choice of set L.