Regularization methods for a problem of analytic continuation (Q413915)

The authors consider the problem of numerical analytic continuation of an analytic function \(f(z)=f(x+iy)\) on the horizontal strip \(\{z\in\mathbb{C}\;|\;\Re(z)\in\mathbb{R},~ 0<\Im(z)\leq y_0)\}\) (for some \(y_0 >0\)), where only \(f(z)|_{y=0}=f(x)\in L^2(\mathbb{R})\) is known approximately. The main goal is to extend this data to the above strip such that NEWLINE\[NEWLINEf(\cdot + iy) \in L^2(\mathbb{R}), \qquad \| f(\cdot + iy_0)\|_p \leq M, NEWLINE\]NEWLINE where \(\|\cdot\|_p\) denotes the norm of the Sobolev space \(H^p(\mathbb{R})\), \(p\geq 0\), and \(M\) is a positive constant.NEWLINENEWLINEA stability estimate for this problem is obtained, and the authors use the generalized Tikhonov regularization method for the corresponding regularization analysis. Moreover, a description of different methods within the framework of regularization theory is given. In the final part of the paper, numerical tests are presented to illustrate the ability of the generalized Tikhonov regularization method when applied to the problem.