Optimal Wavelet Solutions for Ill-posed Elliptic Equations

Abstract

In this paper, a Cauchy problem for two-dimensional Lap lace equation in the strip 0 ≤ × ≤ 1 is considered. This is a classical severely ill-posed problem. Connecting Shannon wavelet bases with a spectral integral of the Hermitian operator, we can obtain a regularized solution. Moreover, some sharp stable estimates between the exact solution and it's approximation is also provided.