Avances en aproximación en el disco unidadEl caso Zernike

Abstract

The objective of this Doctoral Thesis is the study of approximants for functions defined in the unit ball. These approximants are considered using two different approaches: least-squares approximation and uniform approximation. As is well known, least-squares approximation is based on considering inner products defined on the unit ball, and uniform approximation is based on considering the uniform norm, in this case on the unit disk . We give special emphasis to the least-squares approximation based on Zernike orthogonal polynomials, that is, bivariate polynomials which are orthogonal with respect to the Lebesgue measure on the unit disk, due to applications in Optics and Optometry. The first approach is based on approximating functions defined on the �-dimensional ball by studying modifications of the classical inner product (that includes the Zernike polynomials as particular case when the weight function is a constant function) by means of multivariate differential operators such as gradients or Laplacians, the so-called Sobolev inner products in two different ways. First, we deal with the �-dimensional unit ball equipped with an inner product constructed by adding a mass point at the origin to the classical ball inner product applied to the gradients of the functions. We determine an explicit orthogonal polynomial basis, and we study approximation properties of Fourier expansions in terms of this basis. We deduce relations between the partial Fourier sums in terms of the new orthogonal polynomials. We also give an estimate of the approximation error by polynomials of degree at most � in the corresponding Sobolev space, proving that we can approximate a function by using its gradient. The next chapter is devoted to study the orthogonal structure induced by an inner product involving the Laplacians of the functions, an extension of the inner product studied by Xu in 2008 trying to solve the problem posed by Atkinson and Hansen of finding the numerical solution of the nonlinear Poisson equation with zero boundary conditions on the d-dimensional unit ball. We analyze the orthogonal polynomials associated with this new inner product, proving that they satisfy a fourth-order partial differential equation. We also study the approximation properties of the Fourier sums with respect to these orthogonal polynomials and we estimate the error of simultaneous approximation of a function, its partial derivatives, and its Laplacian. In both cases, numerical examples are given to illustrate the approximation behavior of the Sobolev basis. For the second approach, we construct and study sequences of 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. Several interesting numerical examples are given, comparing approximations using the shifted Bernstein-Stancu and the Bernstein-type operator on disk quadrants.