A Modified Tikhonov Regularization for Stable Analytic Continuation

This paper is devoted to a new regularization method for solving the numerical analytic continuation of an analytic function $f(z)=f(x+iy)$ on a strip domain $\Omega^+=\{z=x+iy\in \mathbb{C}\mid x\in \mathbb{R}, 0<y<y_0\}$, where the data is given approximately only on the line $y=0$. This problem is severely ill-posed and has important practical applications. The theoretical optimal error bound for the problem is proved which is independent of the selected regularization methods. A modified Tikhonov regularization method with asymptotic order optimal error estimates is proposed. This method can be numerically implemented easily by the fast Fourier transform. Some numerical examples are provided and a comparison with a Fourier regularization method is given, which show the modified Tikhonov method works very well.