## View abstract

### 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

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.