On the diametral dimension of weighted spaces of analytic germs (Q2809356)

The author continues his investigation of weighted spaces of analytic germs \( \mathcal H_v(\mathbb R) \) from [Ann. Pol. Math. 106, 223--243 (2012; Zbl 1267.46038)]. There, it has been shown that \( \mathcal H_v(\mathbb R) \) is isomorphic to some \( \Lambda_0(\alpha_n)'_b \), i.\,e., to the strong dual of some power series space of finite type. In the present paper, the sequence \( (\alpha_n)_n \) is determined in terms of~\(v\). As an example, it is shown that the space of test functions for the modified Fourier hyperfunctions on~\(\mathbb R\) is isomorphic, as a topological vector space, to \( \Lambda_0(n/\log(n))'_b \). In particular, it is not isomorphic to the space of test functions for the Fourier hyperfunctions on~\(\mathbb R\).NEWLINENEWLINEFor the proof, the diametral dimension of \( \mathcal H_v(\mathbb R) \) is determined, using inheritance properties of the diametral dimension and an explicit embedding of \( \mathcal H_v^\infty(\mathbb D) \) into \( \mathcal H_v^\infty(\mathbb R) \).