Generalized Hilbert and Carleman boundary value problems for functions analytic in a simply connected domain

Abstract

Comparing the results of the second and third chapters, the reader has had an opportunity to be convinced that binomial boundary value problems for functions piecewise analytic in the complex plane and for functions analytic in a simply connected domain are essentially different. These distinctions also manifest themselves in the theory of corresponding polynomial boundary value problems. We explain now one of the most important distinctions. The solutions of a polynomial boundary value problem for a piecewise analytic function are represented in the form of integrals of Cauchy type. By virtue of this representation, the mentioned boundary value problem is equivalent (under the condition Φ−(∞) = 0) to the corresponding characteristic singular integral equation and, that is exactly why the Noether theory of such boundary value problems follows directly, as a particular case, from the Noether theory of the corresponding singular integral operator. For the case of a polynomial boundary value problem for functions analytic in a simply connected domain (finite or infinite), a more complicated situation arises. Integral representations for functions analytic in a domain contain some constants. Therefore the right-hand sides of the integral equations corresponding to boundary value problems depend on these constants. If the constants are uniquely determined, then the boundary value problem and the corresponding singular integral equation are equivalent. However, in this case, for the equivalence of the corresponding allied boundary value problem and the allied singular integral equation, some conditions must be satisfied. If the constants remain arbitrary, then the boundary value problem and singular integral equation are not equivalent but, on the contrary, the allied boundary value problem and the allied singular integral equation are equivalent. A sharp investigation shows that the index of a boundary value problem exceeds the index of the corresponding integral equation by one or two.