Bases in spaces of ultradifferentiable functions with compact support
Let \(I\subset {\mathbb{R}}\) be a compact interval and let \(D_{(M_ p)}(I)\) (and \(D_{\{M_ p\}}(I))\) be the spaces of ultradifferentiable functions with compact support in I of Beurling type (and of Roumieu type, resp.) in the sense of H. Komatsu. It is shown, that \(D_{(M_ p)}(I)\) (and \(D_{\{M_ p\}}(I)_ b')\) are linear topologically isomorphic to a power series space of infinite type (and of finite type, resp.), if \((M_ p)\) satisfies the conditions (M1), (M2) and (M3) of H. Komatsu. The proof is based on Pelczynski's trick: It is shown, that the function spaces and the corresponding sequence spaces contain each other as complemented subspaces. These projections are explicitely given, using certain optimal cut off functions. As an application it is shown, that the problems of E. Borel and H. Whitney may be solved by means of a linear and continuous operator for the ultradifferentiable functions of Beurling type, if \((M_ p)\) satisfies (M1), (M2) and (M3).
spaces of ultradifferentiable functions with compact support
ultradifferentiable functions of Beurling type
scientific article; zbMATH DE number 4019767
scientific article