An analytical solution is presented that reconstructs residual stress field from limited and incomplete data. The inverse problem of reconstructing residual stresses is solved using an appropriate form of the airy stress function. This function is chosen to satisfy the stress equilibrium equations together with the boundary conditions for a domain within a convex polygon. The analytical solution is demonstrated by developing a reference solution from which selected “measurement” points are used. An artificial error is then randomly added to “measurement” points for studying the stability of the reconstruction method utilizing Tikhonov-Morozov regularization technique. It is found that there is an excellent agreement between the model prediction and limited set of residual stress data in the sense of least-square approximation.