Abelian and Tauberian results for the fractional Hankel transform in Zemanian-type spaces

TL;DR

The paper addresses whether the quasiasymptotic behavior of a distribution is preserved under the fractional Hankel transform (FrHT) within a refined Zemanian framework. It resolves this by constructing the Montel space $\mathcal{K}_{-1/2}(\mathbb{R}_+)$ as a projective limit of $\mathcal{K}_{-1/2,\mu}(\mathbb{R}_+)$ and extending the FrHT to the dual $\mathcal{K}'_{-1/2}(\mathbb{R}_+)$, while introducing $\mathcal{B}_{-1/2}(\mathbb{R}_+)$ for Tauberian analysis. The main results include an Abelian-type theorem showing that $f$ with quasi-asymptotics of degree $m$ implies a corresponding quasi-asymptotics for $H_{\mu_0}^{\alpha}f$, and a Tauberian theorem that recovers $f$'s quasi-asymptotics from FrHT limits, all uniform in $\mu_0$. These contributions provide a rigorous, uniform, and largest-distribution-class framework for FrHT asymptotics with potential applications in wave propagation and related areas. The work thus advances distribution theory and transform analysis by linking refined function spaces, Montel properties, and asymptotic behavior under FrHT.

Abstract

In this paper, we first present an Abelian-type theorem for the fractional Hankel transform (FrHT) within Zemanian generalized function spaces. To prove this, we show that these spaces have the Montel property. Next, we construct a new Zemanian-type space as a projective limit of suitable Banach spaces. Its dual is the largest known distribution space admitting the FrHT. Finally, within this extended setting, we establish new Abelian and Tauberian-type results for the FrHT.

Theorems & Definitions (16)

• Example 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Remark 2.5
• Theorem 3.1
• proof
• Lemma 4.1