Polynomial fitting and interpolation on circular sections

We construct Weakly Admissible polynomial Meshes (WAMs) on circular sections, such as symmetric and asymmetric circular sectors, circular segments, zones, lenses and lunes. The construction resorts to recent results on subperiodic trigonometric interpolation. The paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

Near optimal Tchakaloff meshes for compact sets with Markov exponent 2

By a discrete version of Tchakaloff Theorem on positive quadrature formulas, we prove that any real multidimensional compact set admitting a Markov polynomial inequality with exponent 2 possesses a near optimal polynomial mesh. This improves for example previous results on general convex bodies and starlike bodies with Lipschitz boundary, being applicable to any compact set satisfying a uniform interior cone condition. We also discuss two algorithmic approaches for the computation of near optimal Tchakaloff meshes in low dimension

Polynomial Meshes: Computation and Approximation

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs

Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting

Stability inequalities for Lebesgue constants via Markov-like inequalities

We prove that L^infty-norming sets for finite-dimensional multivariatefunction spaces on compact sets are stable under small perturbations. This implies stability of interpolation operator norms (Lebesgue constants), in spaces of algebraic and trigonometric polynomials

Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies

We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in R\u2c6d, and by a tangential Markov inequality via an estimate of the Dubiner distance on smooth convex bodies. These allow to compute a (1 12eps)-approximation to the minimum of any polynomial of degree not exceeding n by O((n/sqrt(eps))\u2c6(\u3b1d)) samples, with \u3b1= 2 in the general case, and \u3b1= 1 in the smooth case. Such constructions are based on three cornerstones of convex geometry, Bieberbach volume inequality and Leichtweiss inequality on the affine breadth eccentricity, and the Rolling Ball Theorem, respectively

Polynomial approximation and quadrature on geographic rectangles

Using some recent results on subperiodic trigonometric interpolation and quadrature, and the theory of admissible meshes for multivariate polynomial approximation, we study product Gaussian quadrature, hyperinterpolation and interpolation on some regions of dS,d ≥ 2. Such regions include caps, zones, slices and more generally spherical rectangles defined on S2 by longitude and (co)latitude (geographic rectangles). We provide the corresponding Matlab codes and discuss several numerical examples on S