Let H (G) be a space of analytic functions of one variable in simply-connected domain G of the complex plane. It is known that a linear complex convolution operator is generated by a one-variable analytic function, a multivalued one in general. A known problem when all such functions are single-valued is solved. As it turned out, the solution to the problem is connected with the geometry of G domain. Set s(G) with property s(G)+G ⊆ G is termed residue of G domain. A class of simply connected regions whose residue is a connected set is described. Let the linear operator be continuous in function space, analytical in simply-connected domain G, and let it commute with differentiation. Then it can be reduced to a complex convolution operator. It is proved that the function generating such an operator will always be single-valued for regions with a connected residue. When the residue of region G is not connected, there is always a complex convolution operator with a multivalued function generating a kernel.
residue of region, operator commuting with operator of differentiation, kernel of operator
Введение. Рассматриваемые в статье задачи входят в направление исследований, представленное работами [1‒7]. Пусть G - односвязная область в комплексной плоскости C , и последовательность ограниченных расширяющихся областей 
с кусочно-гладкой границей исчерпывает G. H (G) - пространство Фреше аналитических в G функций с топологией равномерной сходимости на компактах.
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