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Abelian and Tauberian Results for the Fractional Hankel Transform of Generalized Functions

Sanja Atanasova, Smiljana Jakšić, Snježana Maksimović, Stevan Pilipović

TL;DR

The paper addresses the problem of understanding quasiasymptotics for generalized functions under the fractional Hankel transform (FrHT) by developing a Tauberian framework that links FrHT asymptotics to distributional quasiasymptotics within the Zemanian-type space. It constructs the appropriate distribution spaces 𝒦_μ(R_+) and their duals, then analyzes FrHT on these spaces, establishing both Abelian-type continuity results and Tauberian-type conclusions. The authors prove a Tauberian theorem that connects the existence and growth of FrHT limits to quasi-asymptotics at 0^+ in 𝒦′μ(R_+) and derive initial and final value theorems for FrHT on distribution spaces, providing explicit boundary-behavior formulas. These results advance the understanding of FrHT in distribution theory and have potential applications in wave propagation, optics, and related boundary-value problems where fractional Hankel-type kernels arise.

Abstract

This paper aims to explore the quasiasymptotic behavior of distributions through the fractional Hankel transform. We present Tauberian result that connects the asymptotic behavior of generalized functions in the Zemanian space with the asymptotics of their fractional Hankel transform. Additionally, we establish both the initial and final value theorems for the fractional Hankel transform of distributions.

Abelian and Tauberian Results for the Fractional Hankel Transform of Generalized Functions

TL;DR

The paper addresses the problem of understanding quasiasymptotics for generalized functions under the fractional Hankel transform (FrHT) by developing a Tauberian framework that links FrHT asymptotics to distributional quasiasymptotics within the Zemanian-type space. It constructs the appropriate distribution spaces 𝒦_μ(R_+) and their duals, then analyzes FrHT on these spaces, establishing both Abelian-type continuity results and Tauberian-type conclusions. The authors prove a Tauberian theorem that connects the existence and growth of FrHT limits to quasi-asymptotics at 0^+ in 𝒦′μ(R_+) and derive initial and final value theorems for FrHT on distribution spaces, providing explicit boundary-behavior formulas. These results advance the understanding of FrHT in distribution theory and have potential applications in wave propagation, optics, and related boundary-value problems where fractional Hankel-type kernels arise.

Abstract

This paper aims to explore the quasiasymptotic behavior of distributions through the fractional Hankel transform. We present Tauberian result that connects the asymptotic behavior of generalized functions in the Zemanian space with the asymptotics of their fractional Hankel transform. Additionally, we establish both the initial and final value theorems for the fractional Hankel transform of distributions.
Paper Structure (12 sections, 7 theorems, 46 equations)

This paper contains 12 sections, 7 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Space $\mathcal{K}_{\mu}(\mathbb R_+), \mu\geq -1/2$
  3. Fractional Hankel transform
  4. The quasiasymptotic behavior and boundedness of distributions
  5. Tauberian-type results
  6. Initial and final value theorems for the FrHT on distribution spaces
  7. An initial value theorem for the FrHT
  8. A final value theorem for the FrHT
  9. Declaration
  10. Conflict of interest
  11. Data availability
  12. Acknowledgment

Key Result

Theorem 4.5

Let $f\in \mathcal{K}_{\mu}'(\mathbb R_+)$ and (ex const) hold for $L$, and let the distribution $f$ is quasi-asymptotically bounded of degree $m\in\mathbb R$ at the origin with respect to $L$ in $\mathcal{K}_{\mu}'(\mathbb R_+)$ , i.e., for every $\varphi\in\mathcal{K}_{\mu}(\mathbb R_+)$, where $C_{\varphi} > 0$ depends on $\varphi$. Then the FrHT $H^\alpha_{\mu}(f)$ ($\mu$ is fixed), is bounde

Theorems & Definitions (19)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Remark 4.4
  • Theorem 4.5
  • proof
  • Theorem 4.6
  • proof
  • Lemma 5.1
  • Remark 5.2
  • ...and 9 more