A Survey on q-Polynomials and their Orthogonality Properties

In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson polynomials (AW)) for almost any complex value of the parameters and for all non-negative integers degrees. We state the degenerate version of Favard's theorem, which is one of the keys of the paper, that allow us to extend the orthogonality properties valid up to some integer degree N to Sobolev type orthogonality properties. We also present, following an analogous process that applied in [16], tables with the factorization and the discrete Sobolev-type orthogonality property for those families which satisfy a finite orthogonality property, i.e. it consists in sum of finite number of masspoints, such as q-Racah (qR), q-Hahn (qH), dual q-Hahn (dqH), and q-Krawtchouk polynomials (qK), among others. -- [16] R. S. Costas-Santos and J. F. Sanchez-Lara. Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comp. Appl. Math., 225(2) (2009), 440-451Comment: 3 Figures, 3 tables, in a 22 pages manuscrip

The CMV bispectral problem

A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear differential operator of arbitrary order. Actually, we prove that this is the case even if such an eigenfunction condition is imposed up to finitely many orthogonal Laurent polynomials.Comment: 25 pages, final version, to appear in International Mathematics Research Notice

Numerical integration for high order pyramidal finite elements

We examine the effect of numerical integration on the convergence of high order pyramidal finite element methods. Rational functions are indispensable to the construction of pyramidal interpolants so the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include rational functions and show that despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.Comment: 28 page

On the lack of compactness on stratified Lie groups

In $\mathbb{R}^d$, the characterization of the \mbox{lack of compactness of the continuous Sobolev injection $\mathring{H}^s \hookrightarrow L^p$}, with $\displaystyle{\frac{s}{d} + \frac{1}{p} = \frac{1}{2}}$ and $\displaystyle{

Polynomials of least deviation from zero in Sobolev p-Norm

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p-norm (1<p<∞) for the case p=1. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p-norm. The asymptotic distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that the n-th polynomial of least deviation has at least n−d∗ zeros on the convex hull of the support of the measure, where d∗ denotes the number of terms in the discrete part.The research of H. Pijeira-Cabrera was partially supported by Ministry of Science, Innovation and Universities of Spain, under grant PGC2018-096504-B-C33. Funding Open Access funding provided by Universidad Carlos III de Madrid thanks to the CRUE-CSIC 2021 agreement with Springer Nature

A matrix algorithm towards solving the moment problem of Sobolev type

10 pages, no figures.-- MSC2000 code: 65F30.MR#: MR1832494 (2002g:44005)Zbl#: Zbl 0980.65049We propose a matrix algorithm which is the first step towards considering a given matrix as a moment matrix of Sobolev type in the diagonal form of an arbitrary (not necessarily finite) order on the real line or the unit circle. This continues our recent work [Proc. Amer. Math. Soc., 128 (2000) 2309] on the moment problem of Sobolev type and gives an alternative approach to what is in [J. Approx. Theory 100 (1999) 364].This research was initiated within the framework of scientific and technical cooperation between Spain and Poland supported by the Ministry of Foreign Affairs of Spain and the Committee of Scientific Research (KBN) of Poland, grant 07/R98. The work of the first author (F.M.) was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-C03-01.Publicad