## 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

#### UNLP & IAM-CONICET , Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc429616b3b68024ccc5162c77f4b3aa4').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc429616b3b68024ccc5162c77f4b3aa4 = 'j&#97;&#97;nt&#101;z&#97;n&#97;.d&#111;cs' + '&#64;'; addyc429616b3b68024ccc5162c77f4b3aa4 = addyc429616b3b68024ccc5162c77f4b3aa4 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textc429616b3b68024ccc5162c77f4b3aa4 = 'j&#97;&#97;nt&#101;z&#97;n&#97;.d&#111;cs' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakc429616b3b68024ccc5162c77f4b3aa4').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc429616b3b68024ccc5162c77f4b3aa4 + '\'>'+addy_textc429616b3b68024ccc5162c77f4b3aa4+'<\/a>';

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).