On extension to Fourier transforms

TL;DR

The paper provides estimates for extension operators from bounded functions on finite sets to Fourier transform spaces in LCA groups, showing limitations for infinite closed subsets.

Contribution

It extends previous results by providing new bounds for extension operators and generalizes Graham's result to infinite closed subsets.

Findings

No bounded linear extension operator exists for infinite closed subsets

Established bounds for extension operators from $l^ abla$ to $A( abla)$

Generalized Graham's result to broader classes of sets

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 $K$.