Almost periodic ultradistributions of Beurling and of Roumieu type (Q2719006)

To extend results of \textit{I. Cioranescu} [Proc. Am. Math. Soc. 116, No. 1, 127-134 (1992; Zbl 0765.46021)] on almost periodic ultradistributions on the real line, the author introduces for a given weight function \(\omega\) the bounded ultradistributions of Beurling (resp. Roumieu) type as \({\mathcal D}_{L^1,(\omega)}'(\mathbb{R}^N)\) (resp. \({\mathcal D}_{L^1,\{\omega\}}'(\mathbb{R}^N)\)), where NEWLINE\[NEWLINE\begin{aligned} {\mathcal D}_{L^1,(\omega)}(\mathbb{R}^N) & =\{f\in{\mathcal D}(\mathbb{R}^N): \forall\lambda> 0:\|f\|_\lambda< \infty\},\\ {\mathcal D}_{L^1,\{\omega\}}(\mathbb{R}^N) &= \{f\in{\mathcal D}(\mathbb{R}^N): \exists\lambda> 0:\|f\|_\lambda< \infty\},\\ \|f\|_\lambda &:= \sup_{\alpha\in \mathbb{N}^N_0}\|f^{(\alpha)}\|_{L^1(\mathbb{R}^N)} \exp\Biggl(-\lambda\varphi^*\Biggl({|\alpha|\over \lambda}\Biggr)\Biggr),\end{aligned}NEWLINE\]NEWLINE and where \(\varphi^*\) is the Young conjugate of the convex function \(\varphi(t):= \omega(e^t)\).NEWLINENEWLINENEWLINEA bounded ultradistribution \(T\) is called almost periodic, if \(T\) is the \(\beta({\mathcal D}_{L^1,*}'(\mathbb{R}^N)\), \({\mathcal D}_{L^1,*}(\mathbb{R}^N))\)-limit of a sequence of trigonometric polynomials. Various characterizations of almost periodic ultradistributions are derived and it is investigated how an almost periodic ultradistribution in one variable can be obtained from its spectrum and its Fourier coefficients.