Examinando por Autor "Recarte, Marlon J."

Mostrando 1 - 3 de 3

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.
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.