Regularization of a Class of Summary Equations

Abstract

Let D be an arbitrary quadrangle with boundary $\Gamma$. We consider a four-element linear summary equation. The solution is sought in the class of functions which are holomorphic outside D and vanish at infinity. The boundary values satisfy the Hölder condition on any compact set which does not contain the vertices. At the vertices, singularities at most of logarithmic order are allowed. The coefficients of the equation are holomorphic in D and their boundary values satisfy the Hölder condition on $\Gamma$. The free term satisfies the same conditions. The solution is sought in the form of the Cauchy type integral over $\Gamma$ with unknown density. To regularize the obtained functional equation, we use the Carleman problem. Previously, a Carleman shift is introduced on $\Gamma$; it transfers each side to itself and reverses orientation; the midpoints of the sides are fixed under the shift. We indicate some applications of this summary equation to the problem of moments for entire functions of exponential type.