The Fourier regularization for solving the Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation in an infinite “strip” is considered. The Cauchy data are at the boundary x = 0 given in an approximate manner and the solution is sought in the region { ( x , y ) | 0 < x ⩽ 1 , y ∈ R n , n ⩾ 1 } . This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. In this paper we use the Fourier regularization method to solve the problem. The method is independent of the interval length and wave number. Some sharp error estimates between the exact solution and its regularization approximation are given and numerical examples show that the method works effectively.