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We establish simultaneous approximation properties of weighted first-order Sobolev orthogonal projectors onto spaces of polynomials of bounded total degree in the Euclidean unit ball. The simultaneity is in the sense that we provide bounds for the projection error in both a weighted $\mathrm{L}^2$ norm and a weighted $\mathrm{H}^1$ seminorm, both involving the same weight of the generalized Gegenbauer type $x \mapsto (1-\lVert x \rVert^2)^\alpha$, $\alpha > -1$. The Sobolev orthogonal projectors producing the approximations are with respect to an alternative yet equivalent inner product for the corresponding uniformly weighted $\mathrm{H}^1$ space. In order to obtain our approximation bounds, we study the orthogonal polynomial structure of this alternative Sobolev inner product obtaining, among other results, a characterization of its orthogonal polynomials as solutions of certain Sturm-Liouville problems. We do not rely on any particular basis of orthogonal polynomials, which allows for a streamlined and dimension-independent exposition., Comment: 24 pages

We study approximation properties of weighted $\mathrm{L}^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the reflection-invariant form $(1-\lVert x \rVert^2)^\alpha \prod_{i=1}^d \lvert x_i \rvert^{\gamma_i}$, $\alpha, \gamma_1, \dots, \gamma_d > -1$. Said properties are measured in Dunkl-Sobolev-type norms in which the same weighted $\mathrm{L}^2$ norm is used to control all the involved differential-difference Dunkl operators, such as those appearing in the Sturm-Liouville characterization of similarly weighted $\mathrm{L}^2$-orthogonal polynomials, as opposed to the partial derivatives of Sobolev-type norms. The method of proof relies on spaces instead of bases of orthogonal polynomials, which greatly simplifies the exposition., Comment: 27 pages

We study approximation properties of weighted $L^2$-orthogonal projectors onto spaces of polynomials of bounded degree in the Euclidean unit ball, where the weight is of the generalized Gegenbauer form $x \mapsto (1-\|x\|^2)^\alpha$, $\alpha > -1$. Said properties are measured in Sobolev-type norms in which the same weighted $L^2$ norm is used to control all the involved weak derivatives. The method of proof does not rely on any particular basis of orthogonal polynomials, which allows for a short, streamlined and dimension-independent exposition., Comment: 10 pages; new title, moved about some definitions and the statement of the main result, added remark concerning differently weighted Sobolev spaces, corrected some typos

We study approximation properties of weighted $L^2$-orthogonal projectors onto the space of polynomials of degree less than or equal to $N$ on the unit disk where the weight is of the generalized Gegenbauer form $x \mapsto (1-|x|^2)^\alpha$. The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted $L^2$ norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases and otherwise strongly hinted at by reported numerical tests. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and non-uniformly weighted Sobolev spaces involved, a Markov-type inequality, connection coefficients between Zernike polynomials and relations between the Fourier-Zernike expansions of a function and its derivatives., Comment: 24 pages, 1 figure; new title, new introduction, main result extended and sharpness proved in some cases and suggested by numerical experiments in others, new auxiliary results, updated references

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (J. Non-Newtonian Fluid Mech. 139:153-176, 2006) for the numerical solution of high-dimensional Fokker-Planck equations featuring in Navier-Stokes-Fokker-Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson's equation on a rectangular domain in R^2, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Leli\`evre and Maday (Const. Approx. 30:621-651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173-187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein-Uhlenbeck operator of the kind that appears in Fokker-Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D = D_1 x ... x D_N contained in R^(N d), where each set D_i, i = 1, ..., N, is a bounded open ball in R^d, d = 2, 3., Comment: 45 pages, 2 figures; typos corrected, some proofs simplified, some hypotheses loosened, added and updated references