Examinando por Autor "Marriaga, Misael E."

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Approximation by polynomials in Sobolev spaces associated with classical moment functionals
(Springer, 2023-05-02) García-Ardila, Juan Carlos; Marriaga, Misael E.
Let u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in Hr(u), the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L2 space associated with u. This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N⩽r. We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of En(f(r)), the error of best approximation of f(r) in L2(u).
Approximation via gradients on the ball. The Zernike case
(Elsevier, 2023-10-01) Marriaga, Misael E.; Pérez, Teresa E.; Piñar, Miguel A.; Recarte, Marlon J.
In this work, we deal in a d dimensional unit ball equipped with an inner product constructed by adding a mass point at zero to the classical ball inner product applied to the gradients of the functions. Apart from determining an explicit orthogonal polynomial basis, we study approximation properties of Fourier expansions in terms of this basis. In particular, we deduce relations between the partial Fourier sums in terms of the new orthogonal polynomials and the partial Fourier sums in terms of the classical ball polynomials. We also give an estimate of the approximation error by polynomials of degree at most n in the corresponding Sobolev space, proving that we can approximate a function by using its gradient. Numerical examples are given to illustrate the approximation behavior of the Sobolev basis.
Bernstein-type operators on the unit disk
(Springer Link, 2023-05-30) Recarte, Marlon J.; Marriaga, Misael E.; Pérez, Teresa E.
We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyze the bivariate Bernstein–Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein–Stancu and the Bernstein-type operator on disk quadrants.
Higher-order differential operators having bivariate orthogonal polynomials as eigenfunctions
(Elsevier, 2025-04-14) Marriaga, Misael E.
We introduce a systematic method for constructing higher-order partial differential equations for which bivariate orthogonal polynomials are eigenfunctions. Using the framework of moment functionals, the approach is independent of the orthogonality domain’s geometry, enabling broad applicability across different polynomial families. Applications to classical weight functions on the unit disk and triangle modified by measures defined on lower-dimensional manifolds are presented.
Ladder operators for generalized Zernike or disk polynomials.
(Springer, 2025-04-28) Marriaga, Misael E.
The aim of this work is to report on several ladder operators for generalized Zernike polynomials which are orthogonal polynomials on the unit disk $\mathbf{D}\,=\,\{(x,y)\in \mathbb{R}^2: \; x^2+y^2\leqslant 1\}$ with respect to the weight function $W_{\mu}(x,y)\,=\,(1-x^2-y^2)^{\mu}$ where $\mu>-1$. These polynomials can be expressed in terms of the univariate Jacobi polynomials and, thus, we start by deducing several ladder operators for the Jacobi polynomials. Due to the symmetry of the disk and the weight function $W_{\mu}$, it turns out that it is more convenient to use complex variables $z\,=\, x+iy$ and $\bar{z}\,=\,x-iy$. Indeed, this allows us to systematically use the univariate ladder operators to deduce analogous ones for the complex generalized Zernike polynomials. Some of these univariate and bivariate ladder operators already appear in the literature. However, to the best of our knowledge, the proofs presented here are new. Lastly, we illustrate the use of ladder operators in the study of the orthogonal structure of some Sobolev spaces.
Lecture notes Algebra
(2023-09-05) Marriaga, Misael E.; Stich, Michael
Lecture notes for the course Algebra (Ingeniería Biomédica)
On classical orthogonal polynomials and the Cholesky factorization of a class of Hankel matrices
(World Scientific, 2023-04-11) Marriaga, Misael E.; Vera de Salas, Guillermo; Latorre, Marta; Muñóz Alcázar, Rubén
Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.
On higher order ladder operators for classical orthogonal polynomials
(Springer, 2025-05-12) Marriaga, Misael E.; Martínez, Javier
We analyze the algebraic structure of higher order ladder operators for classical orthogonal polynomials (Hermite, Laguerre, Jacobi, Bessel) in terms of their moment functionals and two first order ladder operators, $J_n^+$ and $J_n^-$. We study operational expressions for these ladder operators in terms of certain integro-differential operators.
On Sobolev orthogonal polynomials on a triangle
(American Mathematical Society, 2023-02) Marriaga, Misael E.
We use the invariance of the triangle T2 = {(x, y) ∈ R2 : 0 < x, y, 1−x−y} under the permutations of {x, y, 1−x−y} to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on T2. These orthogonal polynomials can be constructed from two sequences of univariate orthogonal polynomials. In particular, one of the two univariate sequences of polynomials is orthogonal with respect to a Sobolev inner product and the other is a sequence of classical Jacobi polynomials.
Previous exams with solutions Álgebra
(2023-09-06) Marriaga, Misael E.
Simultaneous approximation via Laplacians on the unit ball
(Springer, 2023-10-13) Marriaga, Misael E.; Pérez, Teresa E.; Recarte, Marlon J.
We study the orthogonal structure on the unit ball $\mathbf{B}^d$ of $\mathbb{R}^d$ with respect to the Sobolev inner products $$ \left\langle f,g\right\rangle_{\Delta} =\lambda\, \mathscr{L}(f,g) + \int_{\mathbf{B}^d}{\Delta[(1-\|x\|^2) f(x)] \, \Delta[(1-\|x\|^2) g(x)]\,dx}, $$ where $\mathscr{L}(f,g) = \int_{\mathbf{S}^{d-1}}f(\xi)\,g(\xi)\,d\sigma(\xi)$ or $\mathscr{L}(f,g) = f(0) g(0)$, $\lambda >0$, $\sigma$ denotes the surface measure on the unit sphere $\mathbf{S}^{d-1}$, and $\Delta$ is the usual Laplacian operator. Our main contribution consists in the study of orthogonal polynomials associated with $\langle \cdot, \cdot \rangle_{\Delta}$, 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 $L^2(\mathbf{B}^d)$ space.
Sobolev orthogonality of polynomial solutions of second-order partial differential equations
(Springer, 2022-12-16) García-Ardila, Juan Carlos; Marriaga, Misael E.
Given a second-order partial differential operator L with nonzero polynomial coefficients of degree at most 2, and a Sobolev bilinear form (P,Q)S=∑i=0N∑j=0i⟨u(i,j),(∂^(i−j)_x∂^j_yP∂^(i−j)_x∂^j_yQ⟩,N⩾0, where u(i,j), 0⩽j⩽i⩽N, are linear functionals defined on the space of bivariate polynomials, we study the orthogonality of the polynomial solutions of the partial differential equation L[p]=λn,mp with respect to (⋅,⋅)S, where λn,m are eigenvalue parameters depending on the total and partial degree of the solutions. We show that the linear functionals in the bilinear form must satisfy Pearson equations related to the coefficients of L. Therefore, we also study solutions of the Pearson equations that can be obtained from univariate moment functionals. In fact, the involved univariate functionals must satisfy Pearson equations in one variable. Moreover, we study polynomial solutions of L[p]=λn,mp obtained from univariate sequences of polynomials satisfying second-order ordinary differential equations.
Study Guide Algebra
(2023-09-06) Marriaga, Misael E.
Study guide Algebra Biomedical Engineering