Conditions for the qualified convergence of finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems in a Hilbert space

We consider finite difference methods and the quasi-reversibility method for solving linear ill-posed Cauchy problems with selfadjoint operators and noise-free initial data in a Hilbert space. We refine the earlier author's results on the convergence rate of the methods under investigation. We establish the sufficient conditions and the necessary conditions, close to one another, for the qualified convergence of these methods in terms of the solution's sourcewise index. We prove that the considered methods cannot converge with the polynomial rate greater than the certain limit, except for the trivial case.