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.
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 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.
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