On extension to Fourier transforms

Abstract

For $n$-point sets $K$ in an LCA group $\Gamma$ we obtain an estimate for the norm of "the best" extension operator from the space $l^\infty(K)$ of bounded functions on $K$ to the space $A(\Gamma)$ of Fourier transforms. As a simple consequence our estimate implies that if $K$ is an infinite closed subset of $\Gamma$ then there does not exist a bounded linear extension operator from the space $C_0(K)$ of continuous functions on $K$ vanishing at infinity to $A(\Gamma)$. The latter result generalizes a result by Graham who considered the case of compact subsets $K$.