Global polynomial optimization by norming sets on sphere and torus

Using the approximation theoretic notion of norming set, we compute(1 12eps)-approximations to the global minimum of arbitrary n-th degree polynomials on the sphere, by discrete minimization on approximately 3.2n^2/eps trigonometric grid points, or 2n^2/eps quasi-uniform points. The same error size is attained by approximately 6.5n^2/eps trigonometric grid points on the torus

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

Abstract Versions of L′Hôpital′s Rule for Holomorphic Functions in the Framework of Complex B-Modules

AbstractAbstract versions of L′Hôpital′s rule are proved for the "ratio" f(z)(g(z))−1, where f : S → X, g : S → A are vector-valued holomorphic functions defined in a region of the complex plane containing S, A being a complex unilal Banach algebra, and X a complex Banach module over A. Both cases, (i) (g(z))−1[formula] 0, and (ii) f(z) [formula] 0, g(z) [formula] 0, as z[formula] α, α being either finite or infinite, are considered when f′(z)(g′(z))−1 has a finite limit. Applications are given to the asymptotics of linear second-order differential equations in Banach algebras

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

Interpolating discrete advection-diffusion propagators at Leja sequences

We propose and analyze the ReLPM (Real Leja Points Method) for evaluating the propagator phi(DeltatB)nu via matrix interpolation polynomials at spectral Leja sequences. Here B is the large, sparse, nonsymmetric matrix arising from stable 2D or 3D finite-difference discretization of linear advection-diffusion equations, and phi(z) is the entire function phi(z) = (e(z) - 1)/z. The corresponding stiff differential system y(t) = By(t) + g,y(0) =y(0), is solved by the exact time marching scheme y(i+1) = y(i) + Deltat(i)phi(Deltat(i)B)(By(i) + g), i = 0, 1,..., where the time-step is controlled simply via the variation percentage of the solution, and can be large. Numerical tests show substantial speed-ups (up to one order of magnitude) with respect to a classical variable step-size Crank-Nicolson solve

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

Bivariate polynomial interpolation on the square at new nodal sets

As known, the problem of choosing ``good'' nodes is a central one in polynomial interpolation. While the problem is essentially solved in one dimension (all good nodal sequences are asymptotically equidistributed with respect to the arc-cosine metric), in several variables it still represents a substantially open question. In this work we consider new nodal sets for bivariate polynomial interpolation on the square. First, we consider fast Leja points for tensor-product interpolation. On the other hand, for classical polynomial interpolation on the square we experiment four families of points which are (asymptotically) equidistributed with respect to the Dubiner metric, which extends to higher dimension the arc-cosine metric. One of them, nicknamed Padua points, gives numerically a Lebesgue constant growing like log square of the degree