Sobolev extremal polynomials and applications

Abstract

The aim of this doctoral dissertation is the study of certain analytic and algebraic properties of polynomials extremal with respect to a Sobolev p-norm and its applications. Problems related to the three cases of Sobolev norms (continuous, discrete and discrete-continuous) are studied independently. In the Continuous case, the extremal polynomials with respect to Sobolev norms of type p are characterized by means of an operator. Results on the location of the zeros of these polynomials as well as its asymptotic behavior are obtained. For extremal polynomials with respect to a discrete Sobolev norm, generated by an inner product, we give a sufficient condition under which the zeros of these polynomials are real and simple The classical Markov's theorem of rational approximation is extended this case of the Sobolev-type orthogonality. In the Discrete-continuous case, the Jacobi-Sobolev space is studied and its completeness is characterized. It is described the cases when the Fourier series converge in Sobolev p-norm.