Regularization of a class of summary equations

Let $ D $ be an arbitrary quadrangle with boundary $\Gamma $. A four-element linear summation equation is considered. The solution is sought in the class of functions that are holomorphic outside $ D $ and disappear at infinity. The boundary values satisfy the Hölder condition on any compact set that does not contain vertices. At the vertices, at most, logarithmic singularities are allowed. Equation coefficients are functions holomorphic in $ D $. Their boundary values satisfy the Hölder condition on $ \Gamma $. The free term satisfies the same conditions. The solution is sought in the form of a Cauchy-type integral over $ \Gamma $ with unknown density. The Carleman problem is used to regularize the resulting functional equation. Previously, a Carleman shift is introduced on $\Gamma $, transferring each side to itself with a change in orientation. The midpoints of the sides are fixed shear points. Applications of this summary equation to the problem of moments for entire functions of exponential type are indicated.