Abstract
In the present paper, we introduce the notion of generalized boundary values for analytic functions on the circle by the method developed in [1–4]. Distributions are treated as linear functionals on some test functions. We introduce the notions of singular integrals with Cauchy kernels (on the circle) and Hilbert kernels for the case in which the density is a distribution. We prove the solvability of a singular integral equation with Hilbert kernel in the class of distributions. We obtain necessary and sufficient conditions for a distribution defined on the circle to be the generalized boundary value of an analytic function and derive an expression for a function analytic outside a circle via its generalized boundary values on the circle.
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