Kernel polynomials from L-orthogonal polynomials

A positive measure ψ defined on [a,b] such that its moments μn=∫a btndψ(t) exist for n=0,±1,±2,⋯, is called a strong positive measure on [a,b]. If 0≤apolynomials {Qn}, defined by ∫a bt-n+sQn(t)dψ(t)=0, s=0,1,⋯,n-1, is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these L-orthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sieved para-orthogonal polynomials on the unit circle

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Nonnegative trigonometric polynomials

An extremal problem for the coefficients of sine polynomials, which are nonnegative in [0,π] , posed and discussed by Rogosinski and Szego is under consideration. An analog of the Fejér-Riesz representation of nonnegative general trigonometric and cosine polynomials is proved for nonnegative sine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szego are obtained explicitly. Associated cosine polynomials k n (θ) are constructed in such a way that { k n (θ) } are summability kernels. Thus, the L p , pointwise and almost everywhere convergence of the corresponding convolutions, is established. © 2002 Springer-Verlag New York Inc.

Multivariate sobolev-type orthogonal polynomials

Multivariate orthogonal polynomials associated with a Sobolev-type inner product, that is, an inner product defined by adding to a measure the evaluation of the gradients in a fixed point, are studied. Orthogonal polynomials and kernel functions associated with this new inner product can be explicitly expressed in terms of those corresponding with the original measure. We apply our results to the particular case of the classical orthogonal polynomials on the unit ball, and we obtain the asymptotics of the kernel functions. © 2011 Universidad de Jaén.

Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Estimates for the Poisson kernel and Hardy spaces on compact manifolds

We study Hardy spaces on the boundary of a smooth open subset or R-n and prove that they can be defined either through the intrinsic maximal function or through Poisson integrals, yielding identical spaces. This extends to any smooth open subset of R-n results already known for the unit ball. As an application, a characterization of the weak boundary values of functions that belong to holomorphic Hardy spaces is given, which implies an F. and M. Riesz type theorem. (C) 2004 Elsevier B.V. All rights reserved.

On the zeros of polynomials: An extension of the Enestrom-Kakeya theorem

This paper presents an extension of the Enestrom-Kakeya theorem concerning the roots of a polynomial that arises from the analysis of the stability of Brown (K, L) methods. The generalization relates to relaxing one of the inequalities on the coefficients of the polynomial. Two results concerning the zeros of polynomials will be proved, one of them providing a partial answer to a conjecture by Meneguette (1994)[6]. (C) 2011 Elsevier B.V. All rights reserved.

Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

Let C-n(lambda)(x), n = 0, 1,..., lambda > -1/2, be the ultraspherical (Gegenbauer) polynomials, orthogonal. in (-1, 1) with respect to the weight function (1 - x(2))(lambda-1/2). Denote by X-nk(lambda), k = 1,....,n, the zeros of C-n(lambda)(x) enumerated in decreasing order. In this short note, we prove that, for any n is an element of N, the product (lambda + 1)(3/2)x(n1)(lambda) is a convex function of lambda if lambda greater than or equal to 0. The result is applied to obtain some inequalities for the largest zeros of C-n(lambda)(x). If X-nk(alpha), k = 1,...,n, are the zeros of Laguerre polynomial L-n(alpha)(x), also enumerated in decreasing order, we prove that x(n1)(lambda)/(alpha + 1) is a convex function of alpha for alpha > - 1. (C) 2002 Published by Elsevier B.V. B.V.

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials P.. (a) of degree m, defined by Boros and Moll via[GRAPHICS]approach the lemmiscate {zeta epsilon C: \zeta(2) - 1\ = Hzeta < 0}, as m --> infinity. (C) 2004 Elsevier B.V. All rights reserved.

Real orthogonal polynomials in frequency analysis

We study the use of para-orthogonal polynomials in solving the frequency analysis problem. Through a transformation of Delsarte and Genin, we present an approach for the frequency analysis by using the zeros and Christoffel numbers of polynomials orthogonal on the real line. This leads to a simple and fast algorithm for the estimation of frequencies. We also provide a new method, faster than the Levinson algorithm, for the determination of the reflection coefficients of the corresponding real Szego polynomials from the given moments.

Connection coefficients and zeros of orthogonal polynomials

We discuss an old theorem of Obrechkoff and some of its applications. Some curious historical facts around this theorem are presented. We make an attempt to look at some known results on connection coefficients, zeros and Wronskians of orthogonal polynomials from the perspective of Obrechkoff's theorem. Necessary conditions for the positivity of the connection coefficients of two families of orthogonal polynomials are provided. Inequalities between the kth zero of an orthogonal polynomial p(n)(x) and the largest (smallest) zero of another orthogonal polynomial q(n)(x) are given in terms of the signs of the connection coefficients of the families {p(n)(x)} and {q(n)(x)}, An inequality between the largest zeros of the Jacobi polynomials P-n((a,b)) (x) and P-n((alpha,beta)) (x) is also established. (C) 2001 Elsevier B.V. B.V. All rights reserved.

Monotonicity of zeros of Jacobi polynomials

Denote by x(nk)(alpha, beta), k = 1...., n, the zeros of the Jacobi polynornial P-n((alpha,beta)) (x). It is well known that x(nk)(alpha, beta) are increasing functions of beta and decreasing functions of alpha. In this paper we investigate the question of how fast the functions 1 - x(nk)(alpha, beta) decrease as beta increases. We prove that the products t(nk)(alpha, beta) := f(n)(alpha, beta) (1 - x(nk)(alpha, beta), where f(n)(alpha, beta) = 2n(2) + 2n(alpha + beta + 1) + (alpha + 1)(beta + 1) are already increasing functions of beta and that, for any fixed alpha > - 1, f(n)(alpha, beta) is the asymptotically extremal, with respect to n, function of beta that forces the products t(nk)(alpha, beta) to increase. (c) 2007 Elsevier B.V. All rights reserved.

Zeros of Gegenbauer and Hermite polynomials and connection coefficients

In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.