Duals of Gelfand-Shilov spaces of type $ K\{M_p\} $ for the Hankel transformation

Abstract

For $ \mu \geq-\frac{1}{2} $, and under appropriate conditions on the sequence $ \{M_p\}_{p = 0}^{\infty} $ of weights, the elements, the (weakly, weakly*, strongly) bounded subsets, and the (weakly, weakly*, strongly) convergent sequences in the dual of a space $ \mathcal{K}_\mu $ of type Hankel-$ K\{M_p\} $ can be represented by distributional derivatives of functions and measures in terms of iterated adjoints of the differential operator $ x^{-1} D_x $ and the Bessel operator $ S_\mu = x^{-\mu-\frac{1}{2}} D_x x^{2 \mu+1} D_x x^{-\mu-\frac{1}{2}} $. In this paper, such representations are compiled, and new ones involving adjoints of suitable iterations of the Zemanian differential operator $ N_\mu = $ $ x^{\mu+\frac{1}{2}} D_x x^{-\mu-\frac{1}{2}} $ are proved. Prior to this, new descriptions of the topology of the space $ \mathcal{K}_\mu $ are given in terms of the latter iterations.

Keywords

• bessel operator
• boundedness
• convergence
• distribution
• generalized function
• $ k\{m_p\} $ space
• representation
• test function
• zemanian operator
• zemanian space