571 research outputs found
An elementary approach to polynomial optimization on polynomial meshes
A polynomial mesh on a multivariate compact set or manifold is a sequence of finite norming sets for polynomials whose norming constant is independent of degree. We apply the recently developed theory of polynomial meshes to an elementary discrete approach for polynomial optimization on nonstandard domains, providing a rigorous (over)estimate of the convergence rate. Examples include surface/solid subregions of sphere or torus, such as caps, lenses, lunes, and slices
Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization
We extend the notion of Dubiner distance from algebraic to trigonometric polynomials on subintervals of the period, and we obtain its explicit form by the Szego variant of Videnskii inequality. This allows to improve previous estimates for Chebyshev-like trigonometric norming meshes, and suggests a possible use of such meshes in the framework of multivariate polynomial optimization on regions defined by circular arcs
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
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
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
Caratheodory-Tchakaloff Subsampling
We present a brief survey on the compression of discrete measures by
Caratheodory-Tchakaloff Subsampling, its implementation by Linear or Quadratic
Programming and the application to multivariate polynomial Least Squares. We
also give an algorithm that computes the corresponding Caratheodory-Tchakaloff
(CATCH) points and weights for polynomial spaces on compact sets and manifolds
in 2D and 3D
Dubiner distance and stability of Lebesgue constants
Some elementary inequalities, based on the non-elementary notion of Dubiner distance, show that norming sets for multivariate polynomials are stable under small perturbations in such a distance. As a corollary, we get anew result on the stability of Lebesgue constants
Chebyshev-Dubiner norming webs on starlike polygons
We construct web-shaped polynomial norming meshes on starlike polygons by radial and boundary Chebyshev points, via the approximation theoretic notion of Dubiner distance. As an application, we get a (1 12eps)-approximation to the minimum of an arbitrary polynomial of degree n by O(n\u2c62/eps) sampling points
Compressed sampling inequalities by Tchakaloff's theorem
We show that a discrete version of Tchakaloff\u2019s theorem on the existence of positive algebraic cubature formulas, entails that the information required for multivariate polynomial approximation can be suitably compresse
Trivariate polynomial approximation on Lissajous curves
We study Lissajous curves in the 3-cube, that generate algebraic cubature
formulas on a special family of rank-1 Chebyshev lattices. These formulas are
used to construct trivariate hyperinterpolation polynomials via a single 1-d
Fast Chebyshev Transform (by the Chebfun package), and to compute discrete
extremal sets of Fekete and Leja type for trivariate polynomial interpolation.
Applications could arise in the framework of Lissajous sampling for MPI
(Magnetic Particle Imaging)
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