Extension of functions on finite sets to Fourier transforms
Abstract
Let $\Gamma$ be an LCA group and $A(\Gamma)$ the corresponding Fourier algebra. We show that if $K\subseteq\Gamma$ is an $n$point set, then $\sqrt{n/2}\leq\alpha_\Gamma(K)\leq\sqrt{n}$ where $\alpha_\Gamma(K)$ is the infimum of the norms of all linear extension operators from $C_0(K)$ to $A(\Gamma)$. The lower bound implies that if $K$ is an infinite closed subset of $\Gamma$, then there does not exist a bounded linear extension operator from $C_0(K)$ to $A(\Gamma)$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07092
 Bibcode:
 2021arXiv211007092L
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Classical Analysis and ODEs;
 Primary 43A20;
 43A25;
 43A46;
 47B38