On perturbed Szegő recurrences

The purpose of the present contribution is to investigate the effects of finite modifications of Verblunsky coefficients on Szegő recurrences. More precisely, we study the structural relations and the corresponding C-functions of the orthogonal polynomials with respect to these modifications from the initial ones.The author thanks the referee for careful reading and valuable comments. The author also wishes to express his thanks to F. Marcellán for suggesting the problem during the 1st Joint Conference of the Belgian, Royal Spanish and Luxembourg Mathematical Societies in Liège, Belgium. The research of the author was supported by CNPq Program/Young Talent Attraction, Ministério da Ciência, Tecnologia e Inovação of Brazil, Project 370291/2013-1 and Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012-36732-C03-01

Accelerated Landweber methods based on co-dilated orthogonal polynomials

In this article, we introduce and study accelerated Landweber methods for linear ill-posed problems obtained by an alteration of the coefficients in the three-term recurrence relation of the \u3bd-methods. The residual polynomials of the semi-iterative methods under consideration are linked to a family of co-dilated ultraspherical polynomials. This connection makes it possible to control the decay of the residual polynomials at the origin by means of a dilation parameter. Depending on the data, the approximation error of the \u3bd-methods can be improved by altering this dilation parameter. The convergence order of the new semi-iterative methods turns out to be the same as the convergence order of the original \u3bd-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is computed in every iteration step