arXiv subjects
Publications and source records attributed to Ruymán Cruz-Barroso.
In this contribution, we introduce a mixed interpolation-regression operator for functions defined on certain planar domains. We focus on an ellipse, an annulus and a polygon. An upper bound for the operator is obtained. Cubature formulas for weight functions defined on these domains are studied. The performance of the interpolation-regression methods is illustrated by numerical examples.
A new alternative numerical procedure to the Szegő quadrature formulas for the estimation of integrals with respect to a positive Borel measure $μ$ supported on the unit circle is presented. As in many practical situations, we assume that the values of the integrand $F$ are only known at a finite number of points, which we will assume to be uniformly distributed on the unit circle (although this does not actually constitute a restriction). Our technique consists of obtaining an approximating Laurent polynomial $L$ to $F$ by interpolation in the Hermite sense in a collection of these points that mimic the zeros of a para-orthogonal polynomial with respect to $μ$, and to use the values of $F$ at the remaining nodes to improve the accuracy of the approximation by a process of simultaneous complex regression. Some numerical examples are carried out.
In this paper we consider an appropriate ordering of the Laurent monomials $x^{i}y^{j}$, $i,j \in \mathbb{Z}$ that allows us to study sequences of orthogonal Laurent polynomials of the real variables $x$ and $y$ with respect to a positive Borel measure $μ$ defined on $\mathbb{R}^2$ such that $\{ x=0 \}\cup \{ y=0 \} \not\in \textrm{supp}(μ)$. This ordering is suitable for considering the {\em multiplication plus inverse multiplication operator} on each varibale $\left( x+\frac{1}{x}\right.$ and $\left. y+\frac{1}{y}\right)$, and as a result we obtain five-term recurrence relations, Christoffel-Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one variable case is also presented, along with some applications for future research.
In this paper we illustrate that paraorthogonality on the unit circle $\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on the unit circle as the analogues of the quasi-orthogonal polynomials on $\mathbb{R}$. We analyze the possibilities of preselecting some of its zeros, in order to build positive quadrature formulas with prefixed nodes and maximal domain of validity. These quadrature formulas on the unit circle are illustrated numerically.