A structure theorem for the space \((G^{\alpha}_{\alpha})^{\prime}\) (Q2854018)

Spaces of type \(G_{\alpha}^{\beta}\) (\(\alpha,\beta\geq 1\)) were introduced by \textit{A. J. Duran} [Indag. Math., New Ser. 3, No. 2, 137--151 (1992; Zbl 0777.46022)]. The spaces \(G_{\alpha}^{\alpha}\) (\(\alpha\geq 1\)) are invariant under the Hankel-Clifford transform and their elements can be characterized in terms of their Fourier-Laguerre coefficients, see [\textit{A. J. Duran}, J. Approximation Theory 74, No. 3, 280--300 (1993; Zbl 0789.41018)].NEWLINENEWLINEThe author of this paper gives a structure theorem for the dual space \((G_{\alpha}^{\alpha})'\), \(\alpha\geq 1\), as follows:NEWLINENEWLINE\( \mathcal{F}\) is an element of \((G_{\alpha}^{\alpha})'\) if and only if there exist a sequence \(\{b_m\}\subset (0,\infty)\) and a continuous bounded function \(f\) on \((0,\infty)\) such that, for every \(d>0\), there exists a constant \(C>0\) satisfying NEWLINE\[NEWLINE \mathcal{F}= \sum_{m=0}^{\infty} b_m (-xD^2-D+\frac{x}{4})^{m}f \quad \text{and} \quad |b_m|\leq\frac{C d^m}{(m!)^{\alpha}}. NEWLINE\]