Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Friday, July 16, 12:45 ~ 13:15 UTC-3

Necessary conditions for interpolation by multivariate polynomials

Jorge Antezana

UNLP & IAM-CONICET , Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $\Omega$ be a smooth, bounded, convex domain in $\mathbb R^n$, and let $\mathcal{P}_k$ be the vector space of of multivariate real polynomials of degree at most $k$. In these spaces we will consider the Hilbert structure given by the $L^2$ norm associated to the Lebesgue measure restricted to $\Omega$.

Consider a sequence $\{\Lambda_k\}_{k\geq 0}$ consisting of finite subsets of $\Omega$. In this talk, we will discuss some necessary geometric conditions for the set $\Lambda_k$ to be interpolating for $\mathcal{P}_k$ and with uniform bounds. Taking as prototype the results about interpolation in spaces of holomorphic functions, the necessary conditions are expressed in terms of an appropriate separation condition, a Carleson condition, and a density condition. On the other hand, in the particular case of the unit ball, we will show that there is not an orthogonal basis of reproducing kernels in the space $\mathcal{P}_k$, when $k$ is big enough.

Joint work with Jordi Marzo (Universidad de Barcelona, España) and Joaquim Ortega Cerdà (Universidad de Barcelona, España).

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