On the null sets of \(2\pi\)-periodic generalized functions (Q2761543)

Let us denote by \({\mathcal D}_{2\pi}\) the set of all \(2\pi\)-periodic, infinitely differentiable functions on \(\mathbb R\) with convergence defined in the following way: a sequence \(\{\varphi_{n}\), \(n\geq 1\}\subset {\mathcal D}_{2\pi}\) converges to the function \(\varphi\) in the space \({\mathcal D}_{2\pi}\) if \(\varphi^{(k)}_{n}\buildrel {\mathbb R}\over{\Rightarrow} \varphi^{(k)}\), \(n\to\infty\) for every \(k\in \mathbb Z\). By \({\mathcal G}_{\{\beta\}}\), \(\beta>0\) we denote the family of all \(2\pi\)-periodic infinitely differentiable functions \(\varphi\) on \(\mathbb R\) such that \(\exists C>0\), \(\exists B>0\), \(\forall x\in {\mathbb R}, \forall m\in {\mathbb Z}_{+}: |\varphi^{(m)}(x)|\leq CB^{m}m^{m\beta}\). The family of all antilinear continuous functionals on \({\mathcal D}_{2\pi}\), \({\mathcal G}_{\{\beta\}}\), \(\beta>0\) with the weak convergence we denote respectively by \({\mathcal D}_{2\pi}'\) and \({\mathcal G}_{\{\beta\}}'\), \(\beta>0\). The authors give three definitions of null set: NEWLINENEWLINENEWLINE1) The generalized function \(F\in {\mathcal D}'_{2\pi}\) \((F\in {\mathcal G}'_{\{\beta\}}\), \(\beta>0)\) is equal to null on the open set \(\Omega\subset{\mathbb R}\), if \(\forall\varphi\in {\mathcal D}_{2\pi} (\forall \varphi\in {\mathcal G}_{\{\beta\}}, \beta>0)\), \(\text{supp }\varphi\subset\Omega:\langle F,\varphi\rangle=0\); NEWLINENEWLINENEWLINE2) The hyperfunction \(F\in {\mathcal G}'_{\{1\}}\) is equal to null on the open set \(\Omega\subset{\mathbb R}\), if its indicatrix \(\widetilde F\) is analytically continued on the arch \(\gamma_{\Omega}=\{e^{i\theta}\), \(\theta\in\Omega\}\); NEWLINENEWLINENEWLINE3) The generalized function \(F\in {\mathcal G}'_{\{\beta\}}\), \(\beta<1\) is equal to null on the open set \(\Omega\subset{\mathbb R}\), if there exists a continuation \(F_{\Omega}\in {\mathcal G}'_{\{\beta\}}({\mathbb R}\setminus \Omega)\) equal to null on \(\Omega\). NEWLINENEWLINENEWLINEThe authors prove that for \(F\in {\mathcal D}_{2\pi}'\) \((F\in {\mathcal G}_{\{\beta\}}'\), \(\beta>1)\) definitions 1 and 3 are equivalent, and for \(F\in {\mathcal G}'_{\{1\}}\) definitions 2 and 3 are equivalent.