Absolutely representing systems of exponentials in the spaces of infinitely- differentiable functions and extendability in the sense of Whitney

Abstract

Let Q be a compactum in $mathbb{R}^p, p geq 1$, such that int Q $neq varnothing$ and $Q = overline{ int Q}$. Denote by $C^{infty}[Q]$ the space of functions from $C^{infty}[intQ]$ uniformly continuous in intQ together with all their partial derivatives. The conditions of the existence of absolutely representing systems of exponentials with purely imaginary exponents in the space $C^{infty}[Q]$ and some of its subspaces of Denjoy--Carleman type are investigated. It is also proved under rather general assumptions that there is no such absolutely representing systems in the space $E(G) = {proj}_{overleftarrow {Qin mathcal{F}G}}E[Q]$ where G is an arbitrary open set in $mathbb{R}^p$, E[Q] is $C^{infty}[Q]$ or its subspace mentioned above and $mathcal{F}_G is the totality of all non-empty compact sets $mathcal{K}$ in G with the property $mathcal{K} = overline{int mathcal{K}}$.