A STRUCTURE THEOREM FOR THE SPACE $({G}^{\alpha}_{\alpha})^{\prime}$

Abstract

For α, β ≥ 1. The space [Formula: see text] is the dual space of [Formula: see text] introduced by Duran in [Laguerre expansions of Gelfand–Shilov spaces, J. Approx. Theory74 (1993)]. We give a structure for the space [Formula: see text], α ≥ 1 as follows: [Formula: see text] if and only if there exist a sequence {bm} ⊂ (0, ∞) and a continuous bounded function f on (0, ∞) such that for every d > 0 there exists a constant C > 0 satisfying [Formula: see text].