On regularization of a summary equation with holomorphic coefficients

F. N. Garifyanov ; E. V. Strezhneva

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 164 (2022) no. 1, pp. 60-67

Résumé

Let $ D $ be a triangle with boundary $ \Gamma = \partial D.$ A six-element linear summary equation in the class of functions that are holomorphic outside $D$ and vanish at infinity is considered. The coefficients of the equation and the free term are holomorphic in $D.$ The solution is sought in the form of a Cauchy-type integral over $\Gamma $ with unknown density. Its boundary value satisfies the Hölder condition on any compact set in $\Gamma$ with no vertices. At the vertices, logarithmic singularities, at most, are allowed. To regularize the equation on $\Gamma$, a piecewise linear Carleman shift is introduced. It maps each side into itself with a change in the orientation. Moreover, at the vertices, it has discontinuity points of the first kind, and the midpoints of the sides are fixed points. The regularization of the equation is carried out, and its equivalence is shown. For this purpose, the theory of the Carleman boundary value problem and the principle of locally conformal gluing are used. Applications to interpolation problems for entire functions of the exponential type are indicated.

Détail

Keywords: summary equation, equivalent regularization, Carleman boundary value problem.