The Coanalytic Problem of Periodic Function Continuation in Spaces with a Weight Having a Singularity at the Boundary

Abstract

It is known that any periodic square-summable function with a power-law weight in a half-strip is uniquely represented as an orthogonal sum of analytic and coanalytic components; therefore, it is natural to regard the coanalytic component as a certain nonanalyticity characteristic of the function. The article considers the problem of finding a periodic function extension so that it would deviate from the weighted Sobolev subspace of analytic functions to a minimal extent (the problem of minimizing the coanalytic deviation). In my previous publication, I considered the problem of extending a function to inside a circle in spaces with a weight that has a singularity at the boundary. Other researchers studied similar coanalytic problems in the weightless case for the unit circle, half-strip, and an arbitrary bounded simply connected domain with a smooth boundary. The coanalytic problem is formulated in the spaces of periodic functions with a weight having a power singularity at the boundary, and the boundary values of the functions are taken from the corresponding Besov space. The known properties of weight spaces, including the direct and inverse theorems about the traces of functions from the classes under consideration have been formulated. Using these properties, a theorem about the existence and uniqueness of the coanalytic problem solution is proved within the principles of the monotone operator theory.