Let \({\mathcal K}={\mathcal K}(n)\) be a set of all the compacts in \({\mathbb{R}}^ n\) with the Hausdorff topology and let \(B_ K\) be a space of all the bounded measurable functions with Fourier transforms supported in \(K\in {\mathcal K}\). The properties of operators F(D) and characteristic prolongation from \(K\in {\mathcal K}(n)\) are discussed.