Comportamento assintótico dos polinômios ortogonais de Sobolev-Jacobi e Sobolev-Laguerre

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Differential orthogonality: Laguerre and Hermite cases with applications

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

Parallel recusive algorithms for inverse discrete Legendre transform and inverse discrete Laguerre transform

This paper presents parallel recursive algorithms for the computation of the inverse discrete Legendre transform (DPT) and the inverse discrete Laguerre transform (IDLT). These recursive algorithms are derived using Clenshaw's recurrence formula, and they are implemented with a set of parallel digital filters with time-varying coefficients.

Continued Fraction Solutions to Hermite's, Legendre's and Laguerre's Differential Equation

The continued fraction method for solving differential equations is illustrated using three famous differential equations used in quantum chemistry.

Laguerre Polynomials, an Introduction

The radial part of the Schrodinger Equation for the H-atom's electron involves Laguerre polynomials, hence this introduction.

Œuvres de Laguerre,

t. 1. Algèbre. Calcul intégral. 1898. xv, 471, [1] p.--t. 2. Géométrie. 1905. [4], 715, [1] p. diagrs.

A day at Laguerre's and other days: being nine sketches,

Printed at the Riverside Press, Cambridge, Mass.

Equisummability Theorems for Laguerre Series

Here we prove results about Riesz summability of classical Laguerre series, locally uniformly or on the Lebesgue set of the function f such that (∫(1 + x)^(mp) |f(x)|^p dx )^(1/p) < ∞, for some p and m satisfying 1 ≤ p ≤ ∞, −∞ < m < ∞.

Group theory and quasiprobability integrals of Wigner functions

The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) approximate to so(2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.

A Review of Operational Matrices and Spectral Techniques for Fractional Calculus

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible considerationof densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended theEuclidean structure of the simplex to a Hilbert space structure of the set of densitieswithin a bounded interval, and van den Boogaart (2005) generalized this to the setof densities bounded by an arbitrary reference density. From the many variations ofthe Hilbert structures available, we work with three cases. For bounded variables, abasis derived from Legendre polynomials is used. For variables with a lower bound, westandardize them with respect to an exponential distribution and express their densitiesas coordinates in a basis derived from Laguerre polynomials. Finally, for unboundedvariables, a normal distribution is used as reference, and coordinates are obtained withrespect to a Hermite-polynomials-based basis.To get the coordinates, several approaches can be considered. A numerical accuracyproblem occurs if one estimates the coordinates directly by using discretized scalarproducts. Thus we propose to use a weighted linear regression approach, where all k-order polynomials are used as predictand variables and weights are proportional to thereference density. Finally, for the case of 2-order Hermite polinomials (normal reference)and 1-order Laguerre polinomials (exponential), one can also derive the coordinatesfrom their relationships to the classical mean and variance.Apart of these theoretical issues, this contribution focuses on the application of thistheory to two main problems in sedimentary geology: the comparison of several grainsize distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock orsediment, like their composition

Traité de l'existence et des attributs de Dieu ; suivi de Lettres sur divers sujets de métaphysique et de religion / par Fénelon,...

[Traité de l'existence de Dieu (français). 1878]