Let P be a hypoelliptic polynomial. We consider classes of ultradifferentiable functions with respect to the iterates of the partial differential operator P(D) and prove that such classes satisfy a Paley-Wiener type theorem. These classes and the corresponding test spaces are nuclear.

In this paper, we study the finite Hankel transformation on spaces of generalized functions by developing a new procedure. We consider two Hankel type integral transformations $h_{\mu }$ and $h_{\mu }^{*}$ connected by the Parseval equation $$\sum _{n=0}^{\infty }\left(h_{\mu }f\right)\left(n\right)\left(h_{\mu }^{*}\varphi \right)\left(n\right)={\int }_0^{1}f\left(x\right)\varphi \left(x\right)dx.$$ A space $S_{\mu }$ of functions and a space $L_{\mu }$ of complex sequences are introduced. $h_{\mu }^{*}$ is an isomorphism from $S_{\mu }$ onto $L_{\mu }$ when $\mu \ge -\frac{1}{2}$. We propose to define the generalized finite Hankel transform $h_{\mu }^{\text{'}}f$ of $f\in S_{\mu }^{\text{'}}$ by $$\langle \left(h_{\mu }^{\text{'}}f\right),{\left(\left(h_{\mu }^{*}\varphi \right)\left(n\right)\right)}_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle ,\text{for}\varphi \in S_{\mu }.$$

Quasi-analyticity theorems of Phragmén-Lindelöf type for holomorphic functions of exponential type on a half plane are stated and proved. Spaces of Laplace distributions (ultradistributions) on ℝ are studied and their boundary value representation is given. A generalization of the Painlevé theorem is proved.

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

Sequence space representations of the spaces DL1,(ω)(RN) and of its dual D'L1,(ω)(RN), the space of bounded ultradistributions of Beurling type, are presented, in case the weight ω is a strong weight.