Explicit representations of spaces of smooth functions and distributions
In his recent paper [Math. Nachr. 287, No. 1, 10--22 (2014; Zbl 1301.46007)], the author constructed a common basis for \(C^\infty(\mathbb{R}^n)\) and \(D(\mathbb{R}^n)\) and several spaces of smooth functions in between (and, by duality, for the corresponding dual spaces). In the present paper, this basis is explicitly calculated. The main analytical result is the presentation of an explicit basis for the space \(\mathcal{E}_0:=\{f\in \mathcal{E}([0,1])\mid \forall k\in\mathbb{N}_0: f^{(k)}(1)=0\}\). This basis is obtained by means of a suitable transform of variables, showing that \(\mathcal{E}_0\) is isomorphic to \(\mathcal{S}_e\), the space of even Schwartz test functions, where even Hermite functions are a basis. Since \(\mathcal{E}_0\) is the building block in the sequence space representation for \(C^\infty(\mathbb{R}^n)\) and \(D(\mathbb{R}^n)\) in [the author, loc. cit.], the basis of \(\mathcal{E}_0\) leads to a common basis for \(C^\infty(\mathbb{R}^n)\) and \(D(\mathbb{R}^n)\) using topological tensor products. These are also used to treat the case of several variables.
scientific article; zbMATH DE number 6383585
scientific article