×

Intersections of non quasi-analytic classes of ultradifferentiable functions. (English) Zbl 1187.46018

For a matrix \({\mathfrak m}=(m_{j,p})_{j\in\mathbb{N},\,p\in\mathbb{N}_0}\) let \(M_{j,p}:= \prod^p_{k=1}m_{j,p}\), \({\mathfrak M}:=(M_{j,p})_{j\in \mathbb{N},\,p\in\mathbb{N}_0}\), and \(M_j: =(M_{j,p})_{p\in\mathbb{N}_0}\). Using the standard notation for ultradifferential classes, the locally convex spaces
\[ {\mathcal E}_{({\mathfrak M})} (\Omega):= \bigcap_{j \in\mathbb{N}}{\mathcal E}^{(M_j)}(\Omega),\quad {\mathcal E}_{({\mathfrak M})} (\Omega):= \bigcap_{j\in \mathbb{N}}{\mathcal E}^{\{M_j\}}(\Omega) \]
as well as \({\mathcal D}_{({\mathfrak M})} (\Omega)\) and \({\mathcal D}_{\{{\mathfrak M}\}}(\Omega)\) are introduced for open sets \(\Omega\) in \(\mathbb{R}^n\). Under mild conditions on \({\mathfrak m}\) (including the non-quasianalyticity of \(M_j\) for each \(j)\) it is shown that \({\mathcal E}_{({\mathfrak M})} (\Omega)={\mathcal E}_{\{{\mathfrak M}\}}(\Omega)\) and \({\mathcal D}_{({\mathfrak M})} (\Omega)= {\mathcal D}_{\{{\mathfrak M}\}}(\Omega)\) holds algebraically. Moreover, \({\mathcal E}_{({\mathfrak M})}(\Omega)\neq{\mathcal E}^{(M)} (\Omega)\) holds in general for all weight sequences \(M\). Various properties of \({\mathcal E}_{({\mathfrak M})}(\Omega)\) and \({\mathcal D}_{({\mathfrak M})}(\Omega)\) are derived. In particular, a sufficient condition of the nuclearity of \({\mathcal E}_{({\mathfrak M})}(\Omega)\) and \({\mathcal D}_{({\mathfrak M})}(\Omega)\) is given.

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A32 Spaces of linear operators; topological tensor products; approximation properties
PDFBibTeX XMLCite