Hyperbolic Fractional-Order Fourier Transforms

• Hyperbolic fractional-order Fourier transformations are generalizations of the classical Fourier transform that replace circular trigonometric functions with hyperbolic analogues.
• They form additive groups with well-defined operator algebras, facilitating spectral analysis in optics, quantum mechanics, and the study of fractional partial differential equations.
• Their adaptability to hyperbolic and fractal geometries underpins advanced imaging, diffraction modeling, and the analysis of anomalous diffusion phenomena.

Hyperbolic fractional-order Fourier transformations generalize the classical fractional Fourier transform by replacing circular trigonometric functions in their integral kernels with hyperbolic analogues, yielding a framework suitable for new classes of propagation and imaging phenomena—especially those involving hyperbolic geometry, fractal dimensionality, or fractional derivatives in physical and mathematical models. These transforms appear as natural extensions in optics, spectral theory, quantum mechanics, and the analysis of partial differential equations on non-Euclidean spaces.

1. Definitions and Mathematical Structure

Hyperbolic fractional-order Fourier transformations (HFrFTs) are defined by substituting the sine, cosine, and associated trigonometric functions in the classical fractional Fourier integral expression with hyperbolic functions such as sinh, cosh, coth, tanh, and csch (Pellat-Finet, 25 Sep 2025, Zhou, 17 Sep 2024). For a real parameter $\beta$ (the fractional order), the first-kind HFrFT on $\mathbb{R}^2$ is given by the integral kernel

$$H_\beta[f](p') = \sqrt{\sinh \beta} \, \exp\left(-\frac{i\pi p'^2\coth \beta}{2}\right) \int_{\mathbb{R}^2} \exp\left(-\frac{i\pi p^2\coth \beta}{2}\right) \exp\left( \frac{2i\pi p\cdot p'}{\sinh \beta} \right) f(p)\, d^2 p.$$

The second-kind transform replaces $\coth \beta$ and $\sinh \beta$ with $\tanh \beta$ and $\cosh \beta$; its kernel is

$$K_\beta[f](p') = \sqrt{\cosh \beta} \exp\left( \frac{i\pi p'^2 \tanh \beta}{2} \right) \int_{\mathbb{R}^2} \exp\left( -\frac{i\pi p^2 \tanh \beta}{2} \right) \exp\left( \frac{2i\pi p\cdot p'}{\cosh \beta} \right) f(p) d^2 p.$$

For general quadratic-phase integral transforms, the HFrFTs correspond to those subgroup representations where the parameter dependence enters through functions such as $\coth(\omega\alpha)$, $\csch(\omega\alpha)$, and $\tanh(\omega\alpha)$ (Zhou, 17 Sep 2024). The classical (circular) fractional Fourier transform is recovered by substituting $\cot\alpha$ and $\csc\alpha$ for their hyperbolic counterparts with the group structure conserved: $H_{\beta'} \circ H_\beta = H_{\beta'+\beta}$ (first kind) (Pellat-Finet, 25 Sep 2025).

2. Group Properties, Composition Laws, and Operator Algebra

Under composition, first-kind HFrFTs form an additive group:

Transform Type	Composition Law	Notes
Circular (classical)	$F_{\alpha'} \circ F_\alpha = F_{\alpha'+\alpha}$	Trigonometric kernels
Hyperbolic (1st kind)	$H_{\beta'} \circ H_\beta = H_{\beta'+\beta}$	Hyperbolic kernels; eigenvalues calculable
Hyperbolic (2nd kind)	$K_{\beta'} \circ K_\beta$ nontrivial	Involves parity and phase factors; noncommutative

Second-kind HFrFTs exhibit more complex behavior, adding noncommutative effects and generating parity operators and additional phase factors upon composition (Pellat-Finet, 25 Sep 2025). Notably, the standard Fourier transform is accessible from the circular family but not directly from the hyperbolic family; $K_0 = i F$ (with $F$ the conventional Fourier transform), indicating nontrivial algebraic relationships between these transformation families.

Weyl pseudo-differential operator theory provides a comprehensive framework: both circular and hyperbolic fractional-order Fourier transforms are instances of Weyl operators with explicit symbols and kernel representations. Their compositions obey the Weyl calculus, permitting analysis by kernel or symbol algebra (Pellat-Finet, 1 Oct 2025).

3. Spectral Theory and Laplacian Diagonalization

In fractional (fractal) spaces, hyperbolic fractional-order Fourier transformations emerge naturally as generalizations of the usual Fourier transform, replacing the Lebesgue measure $dx$ with a fractional measure $d\rho(x) = d^D x \, v_\alpha(x)$, where, for isotropic fractional space, $$v_\alpha(x) = \prod_{\mu=1}^D (x^\mu)^{\alpha - 1} / \Gamma(\alpha)$$ (Calcagni et al., 2012).

The momentum transform is constructed by finding kernel functions—often fractional Bessel functions—that diagonalize a fractional Laplacian operator $K_{\alpha, l}$. For $l = 1/2$, the Laplacian takes the form $K_2 = D^2$ with $$D_\mu = (v_\alpha(x^\mu))^{-1/2} \partial_\mu (v_\alpha(x^\mu))^{1/2}$$. The transform is unitary (satisfies Parseval's identity) if the measure and the kernel are matched accordingly. For hyperbolic spaces, the fractional Laplacian is spectrally defined via the Helgason–Fourier transform, leading to

$$(-\Delta_{H^n})^\alpha u(x) = \mathcal{F}^{-1}\left[ \left( \lambda^2 + \frac{(n-1)^2}{4} \right)^\alpha \hat{u}(\lambda, \theta) \right](x),$$

where the kernel is structurally adapted to the geometry, capturing the hyperbolicity of the underlying space (Li et al., 2022).

4. Analytic Properties and Regularity

The analytic behavior of HFrFTs is governed by the continuity and differentiability of their kernel parameters. For quadratic-phase transforms parametrized by $A(\alpha)$, $B(\alpha)$, $C(\alpha)$, and phase factor $D(\alpha)$, necessary and sufficient conditions for L² continuity, pointwise, and almost-everywhere convergence are established (Zhou, 17 Sep 2024):

• Strong L² continuity: holds if $A(\alpha)$ and $B(\alpha)$ are continuous at $\alpha$.
• Pointwise continuity: for $f \in L^2((1 + t^2)^r dt)$ with $r > 1/2$ or $f \in H^s(\mathbb{R})$ with $s > 1/2$.
• Almost-everywhere convergence: for $f \in H^s(\mathbb{R})$, $s \geq 1/4$.

These results remain valid for the hyperbolic and scaling families, provided their parameters avoid singularities (i.e., domains where csch or coth become unbounded). This establishes a unified analytic theory for all one-parameter subgroups of linear canonical transforms—fractional Fourier, scaling, and hyperbolic types.

5. Applications in Optics, Diffraction, and Imaging

HFrFTs are specifically exploited in the mathematical representation of Fresnel diffraction and coherent optical imaging (Pellat-Finet, 25 Sep 2025, Pellat-Finet, 1 Oct 2025). In scenarios where the diffraction geometry yields a negative parameter $J = (R_A - D)(R_B + D)/[D(D - R_A + R_B)] < 0$, the classical circular transform fails, but the hyperbolic fractional transform succeeds by expressing the propagation integral in “reduced variables” via

$U_B(p') = e^{-i\pi\Phi} H_\beta[u_A](p'),$

with $\coth^2 \beta = -J$.

The composition of two fractional-order transforms enables modeling of imaging through refracting spherical caps. The requirement that the overall transformation be the identity (or parity up to scaling) imposes the constraint that both transforms must be of the same kind (either both circular or both hyperbolic). Imaging laws—including conjugation formulas and radius/lateral magnification—are deduced by manipulating the fractional orders and exploiting the operator algebra. For circular transforms, the radius magnification is given by (Pellat-Finet, 1 Oct 2025):

$m^2 = \left( \frac{n d'}{n' d} \right)^2,$

where $d$ and $d'$ are the algebraic distances from the cap to the object and image, and $n$, $n'$ the respective refractive indices.

6. Extensions to Fractional Spaces and Quantum Theory

The generalization of the Fourier series in fractional dimensional spaces frames the classical cosine-sine basis as “rotated” by a fractional derivative operator $D^\alpha$, resulting in a basis of the form $\cos(n\omega t + (\pi \alpha)/2)$ and $\sin(n\omega t + (\pi \alpha)/2)$ (Dorostkar et al., 2022). The associated rotation matrix mixes the classical Fourier coefficients, paralleling the algebraic structure encountered in HFrFTs.

Quantum mechanical perspectives model fractional Fourier transforms as representations of unitary operators in Hilbert space, extended to families generated by commuting Hermitian operators (Abelian Lie groups), thereby encompassing both standard (circular) and generalized hyperbolic cases. Operator methods such as IWOP facilitate explicit kernel construction (Chen et al., 2013).

7. Connections to Fractional Differential Equations and Propagation Theory

In fractional-hyperbolic systems, the Caputo–Dzhrbashyan fractional derivative modifies the time-evolution operator from the classical exponential $e^{tP}$ to the Mittag–Leffler function $E_\alpha(t^\alpha P(s))$, yielding fractional-order Fourier multipliers in the spatially Fourier-transformed domain (Kochubei, 2012). Fundamental solutions decay exponentially outside the fractional light cone $\{(t,x): |t^{-\alpha} x| < 1\}$, enabling control over propagation even with exponentially growing initial data.

Such approaches interpolate between hyperbolic and parabolic behavior, modeling phenomena such as anomalous diffusion and memory effects in viscoelastic or fractal media. HFrFTs, viewed as spectral transforms adapted to the underlying geometry or fractional derivatives, thus play a central role in the analysis and synthesis of solutions to fractional partial differential equations, both on Euclidean and non-Euclidean (hyperbolic, fractal) spaces.

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In sum, hyperbolic fractional-order Fourier transformations provide a technically rigorous, analytically robust, and geometrically flexible framework for spectral analysis, fractional derivative modeling, and wave propagation in diverse settings, from optics to quantum theory to nonlinear analysis on fractal and hyperbolic spaces. This versatility is underpinned by deep group-theoretic, operator-theoretic, and functional-analytic properties, as established in contemporary research (Calcagni et al., 2012, Zhou, 17 Sep 2024, Pellat-Finet, 25 Sep 2025, Pellat-Finet, 1 Oct 2025, Li et al., 2022, Dorostkar et al., 2022, Kochubei, 2012, Chen et al., 2013).