A Note on Spaces of Absolutely Convergent Fourier Transforms

Let \({\mathcal {F}}f\) be an abolutely convergent Fourier transform on the real line. We extend the following result of K. Karlander to \({\mathbf {R}^{n}}\) for \(n \ge 1\) : Any closed reflexive subspace \(\{ {\mathcal {F}}f \}\) of the space of continuous functions vanishing at infinity is of finite dimension.