Weighted norm inequalities for polynomial expansions associated to some measures with mass points

Fourier series in orthogonal polynomials with respect to a measure $\nu$ on $[-1,1]$ are studied when $\nu$ is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in $[-1,1]$. We prove some weighted norm inequalities for the partial sum operators $S_n$, their maximal operator $S^*$ and the commutator $[M_b, S_n]$, where $M_b$ denotes the operator of pointwise multiplication by b \in \BMO. We also prove some norm inequalities for $S_n$ when $\nu$ is a sum of a Laguerre weight on $\R^+$ and a positive mass on $0$

Estimates of the Lebesgue function of Fourier sums in trigonometric polynomials orthogonal with a weight not belonging to the spaces L (r) (r > 1)

A two-sided pointwise estimate is obtained for the Lebesgue function of Fourier sums in trigonometric polynomials orthogonal with a 2π-periodic weight that differs from the function 1/{pipe} sin(τ/2){pipe} by some factor slowly changing at zero. The weight under consideration does not belong to the space L r for any r > 1. A similar result for polynomials orthogonal on the interval [-1, 1] is obtained in the form of a corollary. © 2012 Pleiades Publishing, Ltd

ASYNPTOTIC PROPERTIES OF ORTHOGONAL POLYNOMIALS AND THEIR DERIVATIVES

The asymptotic properties of the multinomials being orthogonal on the circle and their derivatives of the natural order are investigated. For orthogonal multinomial on the circle the asymptotics in Segeau form being even inner arc not containing the special points of the distribution density has been proved at minimal limitations for the smoothness of the later. For wide classes with special features, the flow evaluations of the Segeau function module, weight analogies of the Markov, Bernstein and Nikolski inequalities and sometimes also the even asymptotic representations of the orthogonal multinomials on the circle and their derivatives have been specified. As a consequence the results about asymptotics of the orthogonal trigonometric polynomials and multinomials being orthogonal on the line (specifically, generalized Jacobian multinomials) have been obtained.Available from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

Asymptotic behaviour of Verblunsky coefficients

Let V (z) = j = 1 m (z - j), h k, h k and | j | = 1, j = 1, ..., m, and consider the polynomials orthogonal with respect to | V | 2 d , n (| V | 2 d ; z), where is a finite positive Borel measure on the unit circle with infinite points in its support, such that the reciprocal of its Szego{combining double acute accent} function has an analytic extension beyond | z | < 1. In this paper we deduce the asymptotic behaviour of their Verblunsky coefficients. By means of this result, an asymptotic representation for these polynomials inside the unit circle is also obtained. © 2006 Elsevier Inc. All rights reserved