Kernel polynomials for the solution of indefinite and ill-posed problems

We introduce a new family of semiiterative schemes for the solution of ill-posed linear equations with selfadjoint and indefinite operators. These schemes avoid the normal equation system and thus benefit directly from the structure of the problem. As input our method requires an enclosing interval of the spectrum of the indefinite operator, based on some a priori knowledge. In particular, for positive operators the schemes are mathematically equivalent to the so-called ν-methods of Brakhage. In a way, they can therefore be seen as appropriate extensions of the ν-methods to the indefinite case. This extension is achieved by substituting the orthogonal polynomials employed by Brakhage in the definition of the ν-methods by appropriate kernel polynomials. We determine the rate of convergence of the new methods and establish their regularizing properties.