Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0 < x 1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0 < x 1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)- norm is also provided. © 2007 Elsevier Inc. All rights reserved