Periodic tempered distributions of Beurling type and periodic ultradifferentiable functions with arbitrary support (Q1751321)

Summary: Let \(\mathcal{S}_\omega'(\mathbb{R})\) be the space of tempered distributions of Beurling type with test function space \(\mathcal{S}_\omega(\mathbb{R})\) and let \(\mathcal{E}_{\omega, p}\) be the space of ultradifferentiable functions with arbitrary support having a period \(p\). We show that \(\mathcal{E}_{\omega, p}\) is generated by \(\mathcal{S}_\omega(\mathbb{R})\). Also, we show that the mapping \(\mathcal{S}_\omega(\mathbb{R}) \rightarrow \mathcal{E}_{\omega, p}\) is linear, onto, and continuous and the mapping \(\mathcal{S}_{\omega, p}'(\mathbb{R}) \rightarrow \mathcal{E}_{\omega, p}'\) is linear and onto where \(\mathcal{S}_{\omega, p}'(\mathbb{R})\) is the subspace of \(\mathcal{S}_\omega'(\mathbb{R})\) having a period \(p\) and \(\mathcal{E}_{\omega, p}'\) is the dual space of \(\mathcal{E}_{\omega, p}\).