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

Subperiodic trigonometric subsampling: A numerical approach

We show that Gauss-Legendre quadrature applied to trigonometric poly- nomials on subintervals of the period can be competitive with subperiodic trigonometric Gaussian quadrature. For example with intervals correspond- ing to few angular degrees, relevant for regional scale models on the earth surface, we see a subsampling ratio of one order of magnitude already at moderate trigonometric degrees

Jacobi norming meshes

We prove by Bernstein inequality that Gauss-Jacobi(-Lobatto) nodes of suitable order are L^inf norming meshes for algebraic polynomials, in a wide range of Jacobi parameters. A similar result holds for trigonometric polynomials on subintervals of the period, by a nonlinear transformation of such nodes and Videnskii inequality

Numerical hyperinterpolation over nonstandard planar regions

We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes

Discrete norming inequalities on sections of sphere, ball and torus

By discrete trigonometric norming inequalities on subintervals of the period, we construct norming meshes with optimal cardinality growth for algebraic polynomials on sections of sphere, ball and torus

Discrete norming inequalities on sections of sphere, ball and torus

By discrete trigonometric norming inequalities on subintervals of the period, we construct norming meshes with optimal cardinality growth for algebraic polynomials on sections of sphere, ball and torus

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

Numerical quadrature on the intersection of planar disks

We provide an algorithm that computes algebraic quadrature formulas with cardinality not exceeding the dimension of the exactness polynomial space, on the intersection of any number of planar disks with arbitrary radius. Applications arise for example in computational optics and in wireless networks analysis. By the inclusion-exclusion principle, we can also compute algebraic formulas for the union of a small number of disks. The algorithm is implemented in Matlab, via subperiodic trigonometric Gaussian quadrature and compression of discrete measures

On "marcov" inequalities

As colleagues and friends we wish to dedicate these pages to Marco Vianello on the occasion of his 60th birthday, which is on October 26, 2021. Marco has made many important contributions to approximation theory and beyond. Here we briefly summarize some of them in the spirit of the occasion