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.

Higher order turän inequalities

The celebrated Turân inequalities P 2 n(x)-P n-x(x)P n+1(x) ≥ 0, x ε[-1,1], n ≥ 1, where P n(x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ 2 n - γ n-1γ n+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the real entire function ψ in the Laguerre-Pölya class, ψ(x) = ∑ ∞ n=0 γ nx n / n!. ©1998 American Mathematical Society.

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.

Estimación de parámetros genéticos para la producción de leche en el día del control empleando un modelo de regresión aleatoria para primeras lactancias en búfalos de la costa norte de Colombia

A total of 5575 monthly test-day yield records from 796 lactations buffaloes first in the north coast of Colombia. The model included random direct additive genetic and permanent environment effects. As fixed effects were included, contemporary groups, and age of cow at calving as covaraible, linear and quadratic effects. Test-day (PLDC) yield was 3.89 ± 1.14 kg. The PLDC ranged from 2.86 kg to 4.26 kg while the highest values towards the middle of lactation. The heritability estimates obtained for PLDC ranged from 0.23 to 0.47. Genetic correlations between PLDC, declining steadily increased the distance between PLDC. Phenotypic variances were higher in the initial PLDC and decreasing towards the end of lactation. The results found in this study indicate that there is a high genetic variability for the PLDC in the population studied using a random regression model.

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

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Sharp Turán inequalities via very hyperbolic polynomials

We present new sharp inequalities for the Maclaurin coefficients of an entire function from the Laguerre-Pólya class. They are obtained by a new technique involving the so-called very hyperbolic polynomials. The results may be considered as extensions of the classical Turán inequalities. © 2010 Elsevier Inc.

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.

Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives

In this paper we analyze the location of the zeros of polynomials orthogonal with respect to the inner product where α >-1, N ≥ 0, and j ∈ N. In particular, we focus our attention on their interlacing properties with respect to the zeros of Laguerre polynomials as well as on the monotonicity of each individual zero in terms of the mass N. Finally, we give necessary and sufficient conditions in terms of N in order for the least zero of any Laguerre-Sobolev-type orthogonal polynomial to be negative. © 2011 American Mathematical Society.

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.

Szego{double acute} and para-orthogonal polynomials on the real line: Zeros and canonical spectral transformations

We study polynomials which satisfy the same recurrence relation as the Szego{double acute} polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szego{double acute} polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szego{double acute} polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szego{double acute} polynomials and polynomials arising from canonical spectral transformations are obtained. © 2012 American Mathematical Society.