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Seminario: “On multivariate “needle” polynomials and their application to norming sets and cubature formulas”

Venerdì 17 Novembre 2017, ore 11:30 - Aula 2BC30 - András Kroó

ARGOMENTI: Seminari

Venerdì 17 Novembre 2017 alle ore 11:30 in Aula 2BC30, András Kroó (Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary) terrà un seminario dal titolo “On multivariate “needle” polynomials and their application to norming sets and cubature formulas”.
Abstract
Univariate needle polynomials $p_n$ of degree $n$ on the interval $[-1, 1]$ attain value $1$ at some $x_0 \in [-1, 1]$ and are “exponentially small” $0 \leq p_n(x) \leq e^{-n\phi(h)}$ whenever $|x| \leq 1, |x - x_0 | > h$ with $\phi(h) \downarrow 0$ being some positive function of $h > 0$ depending on the location of $x_0$. These polynomials which are widely applied in various areas can be viewed as the optimal polynomial Dirac delta functions.
In this talk we shall discuss properties of multivariate needle polynomials. Even in the univariate case there is an essential difference between the rate of decrease of the needle polynomials at the inner and boundary points of the interval. This phenomena becomes more intricate in the multivariate case. We will show how the decrease of the multivariate needle polynomials at the boundary points of convex bodies is related to the geometry of the boundary. This will be accomplished for both ordinary and homogeneous multivariate polynomials. Finally, we shall discuss how properties of needle polynomials can be applied in the study of norming sets (optimal meshes) and cubature formulas.