Multivariate “needle” polynomials with application to norming sets and cubature formulas

Kroó, András (2015) Multivariate “needle” polynomials with application to norming sets and cubature formulas. Acta Mathematica Hungarica, 147 (1). pp. 46-72. ISSN 0236-5294 (print), 1588-2632 (online)

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Abstract

Needle polynomials 0≤pn≤1 of degree n on the interval [−1, 1] attain the value p<inf>n</inf>(x0)=1 at some x0∈[-1,1] and are “exponentially small”, p<inf>n</inf>(x)≤e-nϕ(h), whenever x∈[-1,1]\[-h+x0,h+x0], 0<h≤1 with ϕ(h)↓0 being some positive function depending on h and the location of x<inf>0</inf>. In this paper we intend to give a comprehensive study of multivariate needle polynomials. Even in the univariate case there is an essential difference between the sharpness of the corresponding needles at the inner and boundary points. This phenomenon becomes more intricate in the multivariate case. We will show that the sharpness of multivariate needle polynomials at the boundary points of convex bodies is closely related to the geometry of the boundary at the corresponding points. 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. © 2015, Akadémiai Kiadó, Budapest, Hungary.

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