New Zemanian Type Spaces and the Quasiasymptotics for the Fractional Hankel Transform

We present an Abelian theorem for the fractional Hankel transform (FrHT) on the Montel space $\mathcal{K}_{-1/2}(\RR_+)$, which is designed to overcome the limitations of the Zemanian space $\mathcal K^{\mu}(\R_+)$. The essential part of the paper is a new construction of the basic space $\mathcal{K}_{-1/2}(\RR_+)$, ensuring Montel space properties that guarantee desirable topological features such as the equivalence of weak and strong convergence. Also, we prove the continuity of the FrHT on this Montel space, both in function and distribution settings. The paper concludes with a Tauberian theorem that provides the converse implication showing that under appropriate growth and limit conditions the asymptotic behavior of the original distribution can be deduced from that of its FrHT. For this purpose a new space $\mathcal B_{-1/2}(\RR_+)$ is construct.