Another connection between orthogonal polynomials and L-orthogonal polynomials

We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the class S-3 [0, beta, b]. Examples are given to illustrate the main contribution in this paper. (c) 2006 Elsevier B.V. All rights reserved.

Blumenthal's theorem for Laurent orthogonal polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B-n(x) = (x - beta(n))beta(n-1)(x) - alpha(n)xB(n-2)(x), with positive recurrence coefficients alpha(n+1),beta(n) (n = 1, 2,...). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where alpha(n) --> alpha and beta(n) --> beta and show that the zeros of beta(n) are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials. (C) 2002 Elsevier B.V. (USA).

Szego polynomials and the truncated trigonometric moment problem

We show how Szego polynomials can be used in the theory of truncated trigonometric moment problem.

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions d phi(1)(x) and d phi(2)(x) such that d phi(2)(x) = (I + kx(2))d phi(1)(x). AS applications of these properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.

Chain sequences and symmetric generalized orthogonal polynomials

in this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K-n((lambda.,M,k)) associated with the probability measure dphi(lambda,M,k;x), which is the Gegenbauer measure of parameter lambda + 1 with two additional mass points at +/-k. When k = 1 we obtain information on the polynomials K-n((lambda.,M)) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K-n((lambda,M,k)) in relation to M and k are also given. (C) 2002 Elsevier B.V. B.V. All rights reserved.

Chebyshev-Laurent polynomials and weighted approximation

Let (a, b) subset of (0, infinity) and for any positive integer n, let S-n be the Chebyshev space in [a, b] defined by S-n:= span{x(-n/2+k),k= 0,...,n}. The unique (up to a constant factor) function tau(n) is an element of S-n, which satisfies the orthogonality relation S(a)(b)tau(n)(x)q(x) (x(b - x)(x - a))(-1/2) dx = 0 for any q is an element of Sn-1, is said to be the orthogonal Chebyshev S-n-polynomials. This paper is an attempt to exibit some interesting properties of the orthogonal Chebyshev S-n-polynomials and to demonstrate their importance to the problem of approximation by S-n-polynomials. A simple proof of a Jackson-type theorem is given and the Lagrange interpolation problem by functions from S-n is discussed. It is shown also that tau(n) obeys an extremal property in L-q, 1 less than or equal to q less than or equal to infinity. Natural analogues of some inequalities for algebraic polynomials, which we expect to hold for the S-n-pelynomials, are conjectured.

Szego type polynomials and para-orthogonal polynomials

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Zeros of orthogonal polynomials generated by canonical perturbations of measures

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

SZEGO and PARA-ORTHOGONAL POLYNOMIALS on THE REAL LINE: ZEROS and CANONICAL SPECTRAL TRANSFORMATIONS

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

On a moment problem associated with Chebyshev polynomials

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Squeezed states, generalized Hermite polynomials and pseudo-diffusion equation

We show that the wavefunctions 〈pq; λ|n〈, of the harmonic oscillator in the squeezed state representation, have the generalized Hermite polynomials as their natural orthogonal polynomials. These wavefunctions lead to generalized Poisson Distribution Pn(pq;λ), which satisfy an interesting pseudo-diffusion equation: ∂Pnp,q;λ) ∂λ= 1 4 [ ∂2 ∂p2-( 1 λ2) ∂2 ∂q2]P2(p,q;λ), in which the squeeze parameter λ plays the role of time. Th entropies Sn(λ) have minima at the unsqueezed states (λ=1), which means that squeezing or stretching decreases the correlation between momentum p and position q. © 1992.

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.

Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.

Zeros of Jacobi-Sobolev orthogonal polynomials following non-coherent pair of measures

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)