Regularized solution of the Cauchy problem for the Laplace equation using Meyer wavelets

We consider the Cauchy problem for the Laplace equation in the half plane x > 0 where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 < x < 1. This is a model ill-posed problem since a small perturbation in the initial data leads to large errors in the solution. We use Meyer wavelet transform as a regularization procedure to restore the stability of the solution and show that under certain conditions this regularized solution is convergent to the exact solution when a data error tends to zero.