Multivariate Orthogonal Polynomials and Modified Moment Functionals

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained by adding to the moment functional a finite set of mass points, or by multiplying it times a polynomial of total degree 2, respectively. Orthogonal polynomials associated with modified moment functionals will be studied, as well as the impact of the modification in useful properties of the orthogonal polynomials. Finally, some illustrative examples will be given

Making Transport Safer: V2V-Based Automated Emergency Braking System

An important goal in the field of intelligent transportation systems (ITS) is to provide driving aids aimed at preventing accidents and reducing the number of traffic victims. The commonest traffic accidents in urban areas are due to sudden braking that demands a very fast response on the part of drivers. Attempts to solve this problem have motivated many ITS advances including the detection of the intention of surrounding cars using lasers, radars or cameras. However, this might not be enough to increase safety when there is a danger of collision. Vehicle to vehicle communications are needed to ensure that the other intentions of cars are also available. The article describes the development of a controller to perform an emergency stop via an electro-hydraulic braking system employed on dry asphalt. An original V2V communication scheme based on WiFi cards has been used for broadcasting positioning information to other vehicles. The reliability of the scheme has been theoretically analyzed to estimate its performance when the number of vehicles involved is much higher. This controller has been incorporated into the AUTOPIA program control for automatic cars. The system has been implemented in Citroën C3 Pluriel, and various tests were performed to evaluate its operation

Sales model: a preliminary approach and methodology

This work project aims at supporting managers in explaining and predicting sales of specificproducts based on a devised methodology. Product sales time series were analysed andprocessed in order to select the best model type: explanatory models (through ordinary leastsquares method), univariate models (Box Jenkins methodology) or dynamic models mixing upthe two previous approaches. An automatic procedure to put the methodology in practice wasimplemented using Python, due to the huge amount of product sales to be modelled. Theprocess was tested using data from more than 1500 products from Beiersdorf Lisbon. For thesake of confidentiality, the names of the products were modified. The most accurate models aredescribed and analyzed

Quadratic decomposition of bivariate orthogonal polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Backlund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.publishe

Three term relations for multivariate Uvarov orthogonal polynomials

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite.Agencia Estatal de Investigación | Ref. PID2020-113275GB-I00IMAG-María de Maeztu | Ref. CEX2020-001105-MAgencia Estatal de Investigación | Ref. PGC2018-094932-B-I00Junta de Andalucía | Ref. A-FQM-246-UGR20Universidade de Vigo/CISU

Simultaneous Approximation via Laplacians on the Unit Ball

We study the orthogonal structure on the unit ball Bd of Rd with respect to the Sobolev inner products f, g Δ = λL(f, g) + Bd Δ[(1 − x 2)f(x)] Δ[(1 − x 2)g(x)] dx, where L(f, g) = Sd−1 f(ξ) g(ξ) dσ(ξ) or L(f, g) = f(0)g(0), λ > 0, σ denotes the surface measure on the unit sphere Sd−1, and Δ is the usual Laplacian operator. Our main contribution consists in the study of orthogonal polynomials associated with ·, · Δ, giving their explicit expression in terms of the classical orthogonal polynomials on the unit ball, and proving that they satisfy a fourth-order partial differential equation, extending the well-known property for ball polynomials, since they satisfy a second-order PDE.We also study the approximation properties of the Fourier sums with respect to these orthogonal polynomials and, in particular, we estimate the error of simultaneous approximation of a function, its partial derivatives, and its Laplacian in the L2(Bd) space.Funding for open access publishing: Universidad de Granada/CBUAFunding for open access charge: Universidad de Granad