Fractional Hankel–Bessel Transform

• The fractional Hankel–Bessel transform is a family of integral transforms that introduces a continuous deformation parameter, bridging the identity and classical Hankel transforms.
• It exhibits unitary properties and a semigroup composition rule with an explicit inversion formula, underpinning its use in harmonic analysis and pseudo-differential operator theory.
• The transform extends to specialized function and distribution spaces, facilitating detailed symbolic calculus, asymptotic analysis, and applications in wave propagation.

The fractional Hankel–Bessel transform generalizes the classical Hankel transform by introducing a continuous deformation parameter—often called the "fractional angle"—that interpolates between the identity and the classical Hankel transform. This family of integral transforms incorporates a fractional Fourier-type phase in the kernel, and has applications in generalized harmonic analysis, pseudo-differential operator theory, distribution theory, and quaternionic analysis.

1. Formal Definition and Kernel Structure

Let $\mu > -\frac{1}{2}$ and $\alpha \in \mathbb{R} \setminus \pi\mathbb{Z}$ (or $\alpha \in (0,\pi)$ in some conventions). For $f \in \mathcal{S}(0,\infty)$ or on suitable test/function spaces (see below), the fractional Hankel–Bessel transform is defined as

$$(\mathcal{H}_{\mu,\alpha}f)(x) = \int_0^\infty K_\alpha(x,y) f(y)\,dy$$

where the kernel is given by

$$K_\alpha(x,y) = (x y)^{1/2}\, J_\mu(xy\,\sin \alpha)\; e^{\frac{i}{2}(x^2 + y^2)\cos \alpha}$$

with $J_\mu$ the Bessel function of the first kind. Alternative conventions for the kernel normalization appear in the literature, particularly with regard to the "weight" and overall phase factors (Pasawan, 6 Jan 2026, Atanasova et al., 29 Apr 2025, Atanasova et al., 12 Nov 2025, Mahato, 2018). The kernel reduces to that of the classical Hankel transform for $\alpha = \pi/2$, i.e.,

$$(\mathcal{H}_{\mu} f)(x) = \int_0^\infty (xy)^{1/2} J_\mu(xy) f(y)\,dy$$

while for $\alpha = 0$ or $\alpha = \pi$, it is the identity.

2. Inversion, Unitarity, and Group Properties

The fractional Hankel–Bessel transform $\mathcal{H}_{\mu,\alpha}$ extends uniquely to a unitary operator on $L^2(0,\infty)$, and the inversion formula is explicit: $$(\mathcal{H}_{\mu, \alpha})^{-1} = \mathcal{H}_{\mu, -\alpha}$$ With $\langle \cdot, \cdot\rangle_{L^2}$ denoting the $L^2$ inner product,

$$\langle \mathcal{H}_{\mu, \alpha} f, \mathcal{H}_{\mu, \alpha} g\rangle_{L^2} = \langle f, g\rangle_{L^2}$$

and

$$\| \mathcal{H}_{\mu,\alpha} f\|_{L^2} = \|f\|_{L^2}$$

The family of transforms obeys the composition rule

$$\mathcal{H}_{\mu,\alpha} \circ \mathcal{H}_{\mu,\beta} = \mathcal{H}_{\mu,\alpha+\beta}$$

This semigroup structure is parallel to the behavior of the classical fractional Fourier transform and is foundational for pseudo-differential calculus and symbolic analysis associated with the transform (Pasawan, 6 Jan 2026, Elkachkouri et al., 2020).

3. Function Spaces and Distributional Theory

Zemanian-Type and Montel Spaces

The natural function spaces for the fractional Hankel–Bessel transform are the Zemanian spaces $K^\mu(\mathbb{R}_+)$, consisting of all $C^\infty$-functions on $(0,\infty)$ such that for all $m, k \geq 0$,

$$\gamma_{m,k}^\mu(\phi) = \sup_{x>0}\left|x^{m} (x^{-1}D)^k[x^{-\mu-1/2} \phi(x)]\right| < \infty$$

These are Fréchet spaces, but not Montel; to address this, new projective-limit Montel spaces (e.g., $\mathcal{K}_{-1/2}(\mathbb{R}_+)$) have been constructed, defined by seminorms involving growth and smoothness with respect to appropriately adjusted Bessel weights and derivatives. Their Montel property ensures strong topology and reflexivity—key for Abelian and Tauberian theorems (Atanasova et al., 12 Nov 2025, Atanasova et al., 29 Apr 2025).

The transform extends by duality to spaces of distributions: $K^\mu(\mathbb{R}_+)'$ and the largest known such space $(\mathcal{K}_{-1/2}(\mathbb{R}_+))'$, which contains all compactly supported distributions and is complete, with well-described topologies via seminorms.

Gel'fand–Shilov and Ultradifferentiable Spaces

Continuity of the fractional Hankel–Bessel transform is also investigated on Gel'fand–Shilov type S and spaces of ultradifferentiable functions, with precise control over their behavior under fractional Hankel transforms and associated wavelet transforms (Mahato, 2018).

4. Pseudo-differential Operators and Symbolic Calculus

Pseudo-differential operators $A_{\sigma,\alpha}$ associated with the fractional Hankel–Bessel transform are defined by

$$(A_{\sigma,\alpha}f)(x) = \int_0^\infty K_\alpha(x,y)\,\sigma(x,y)f(y)\,dy$$

where $\sigma$ is a symbol belonging to the “global Shubin-type” class $S_{FH}^m$ described by

$$|\partial_x^k \partial_y^\ell \sigma(x,y)| \leq C_{k,\ell} (1+x^2+y^2)^{(m-k-\ell)/2}$$

for all $x, y > 0$ and permissible derivatives. These symbol classes inherit algebraic and mapping properties (closure under product, differentiation, and decay/growth estimates) resembling the classical Shubin classes, adapted to the fractional Hankel–Bessel geometry.

The kernel $M_{\sigma,\alpha}(x,y)$ of $A_{\sigma,\alpha}$ satisfies decay and regularity properties governed by the order $m$ of the symbol, enabling the development of a pseudo-differential calculus for fractional Hankel operators (Pasawan, 6 Jan 2026).

Boundedness and continuity hold for $A_{\sigma,\alpha}$ on $L^p$ spaces when $\sigma\in S_{FH}^0$, and more generally, these operators map between fractional Hankel–Sobolev spaces $H_{\alpha,\mu}^s$, with boundedness estimates governed by the order of $\sigma$ (Pasawan, 6 Jan 2026).

5. Abelian and Tauberian Theorems for Quasiasymptotics

The Abelian–Tauberian circle for the fractional Hankel–Bessel transform provides precise connections between the quasiasymptotic behavior of distributions (in the sense of Zemanian or Montel-type test spaces) and the corresponding asymptotics of their transforms. Explicitly, for $f \in (\mathcal{K}_{-1/2})'$, a quasiasymptotic of degree $m$ at $0$ of the form

$$f(\varepsilon x) \sim \varepsilon^m L(\varepsilon) u(x) \quad \text{as}\ \varepsilon\to 0^+$$

(where $L(\varepsilon)$ is slowly varying, $u$ homogeneous) yields

$$e^{i c_1 (\xi/\varepsilon)^2/2} H_\mu^\alpha[f](\xi/\varepsilon) \sim C_{\alpha,\mu} c_2^{m+1} \varepsilon^{m+1} L(\varepsilon) H_\mu[u](\xi)$$

in $(\mathcal{K}_{-1/2})'$. Tauberian converses of this statement allow recovery of distributional quasiasymptotics from those of the transform, under mild growth/limit conditions. These results generalize the classical Abelian and Tauberian theorems for Fourier and Hankel transforms to the fractional context (Atanasova et al., 12 Nov 2025, Atanasova et al., 29 Apr 2025, Atanasova et al., 29 Apr 2025).

Initial and final value theorems also hold: for locally integrable $f$ and suitable $\eta$, the asymptotics as $x\to 0$ (resp. $x\to\infty$) of $f(x)$ are recovered from the $\xi\to\infty$ (resp. $\xi\to 0$) behavior of $H_\mu^\alpha[f](\xi)$, accompanied by phase factors and power-law weights dictated by the kernel structure (Atanasova et al., 29 Apr 2025).

6. Quaternionic and Bargmann Extensions

A quaternionic-valued fractional Hankel transform $\mathcal{L}^a_\theta$ is constructed via the hyperholomorphic second Bargmann transform for the slice Bergman space of the second kind: $$(\mathcal{L}^a_\theta \phi)(y) = \frac{y^{a/2}}{(1-\theta)^{a+1}} \int_0^\infty x^{a/2} I_a\left(\frac{2\sqrt{\theta x y}}{1-\theta}\right) e^{-\frac{\theta x + y}{1-\theta}} \phi(x) dx$$ with $I_a$ the modified Bessel function, $\theta \in \mathbb{H}$, $|\theta|<1$ or $|\theta|=1$. When $\theta$ lies in a complex slice and is parametrized as $e^{i\alpha}$, the operator reduces to the classical real fractional Hankel transform. The operator $\mathcal{L}^a_\theta$ is unitary on $L^{2,a}(\mathbb{R}^+)$ for $|\theta|=1$, with the inverse given by $\mathcal{L}^a_{\theta^{-1}}$. The quaternionic Bargmann–versus construction thus generalizes the theory to non-commutative and slice-regular settings (Elkachkouri et al., 2020).

7. Applications and Illustrative Examples

The fractional Hankel–Bessel transform framework admits a wide range of explicit computations and operational rules.

• Delta function: $$H_\mu^\alpha[\delta_a](\xi) = K_{\alpha,\mu}(a,\xi)$$.
• Power-law distributions: For $f(x) = x^\nu$, closed-form transforms in terms of confluent hypergeometric functions may be derived by reducing the computation to the classical Hankel transform as

$$H_\mu^\alpha[x^\nu](\xi) = C_{\alpha,\mu} e^{-i (c_1/2)\,\xi^2} H_\mu[x^\nu e^{-i (c_1/2)x^2}](c_2\xi)$$

where $c_1 = \cot \alpha$, $c_2 = \csc \alpha$ (Atanasova et al., 29 Apr 2025).

• Gaussian: $$f(x) = e^{-\beta x^2} \implies H_\mu^\alpha[f](\xi) = \text{const} \cdot e^{ -\frac{ (\beta + i c_1/2) (c_2^2 \xi^2) }{ 4\beta + 2i c_1 } }$$.

Operational properties extend to the symbolic, convolution, and scaling laws, and the framework accommodates the development of wavelet transforms adapted to the fractional Hankel–Bessel structure (Mahato, 2018). The pseudo-differential calculus enables Sobolev space mapping properties and $L^p$ boundedness for fractional Hankel–Bessel operators (Pasawan, 6 Jan 2026).

The continuity, invertibility, and Abelian–Tauberian theorems on Montel and Zemanian-type test and distribution spaces activate a broad theoretical apparatus that sharpens the analysis in harmonic analysis, PDE, and signal processing, especially the asymptotics of wave propagation and optical models.

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Key references: (Pasawan, 6 Jan 2026, Atanasova et al., 29 Apr 2025, Atanasova et al., 12 Nov 2025, Atanasova et al., 29 Apr 2025, Mahato, 2018, Elkachkouri et al., 2020).