A problem of the best continuation of a periodic function (Q1594005)

Assume that \(f_{0}\) is a complex function of real argument which is \(2\pi\)-periodic. Take \(G=\{(x,y): -\pi<x<\pi,y>0\}\) and \(p>1\). It is known that every function \(f_{0}\in W_{p}^{1-1/p}(-\pi,\pi)\) admits a \(2\pi\)-periodic continuation \(f\) into the upper half-plane \(y>0\) such that it belongs to the Sobolev space \(W_{p}^{1}(G)\). Any function \(f\in L_{p}(G)\) can be written as a sum of its analytic part \(f_{a}\) and coanalitic part \(f-f_{a}\). The author study and gives a solution for the problem of finding of the continuation of \(f_{0}\) having the least coanalytic deviation in the sense of a minimality of \[ \iint_{G}\left[\left|\frac{\partial}{\partial x}(f-f_{a})(z)\right|^{p} +\left|\frac{\partial}{\partial y}(f-f_{a})(z)\right|^{p} +\left|(f-f_{a})(z)\right|^{p}\right]dxdy. \] The author uses a differential mathematical model of the above minimization problem.