Joint Time-Vertex Fractional Fourier Transform

• JFRFT is a separable fractional extension of the joint time-vertex Fourier transform that independently applies fractional operations in time and graph domains.
• It enables effective spatiotemporal analysis for tasks such as filtering, denoising, and sampling, reducing to identity at zero orders and standard Fourier at unit orders.
• The framework supports learning-based optimization and adaptive fractional orders, offering improved energy compaction and enhanced signal recovery over conventional methods.

The Joint Time-Vertex Fractional Fourier Transform (JFRFT), also written as JFRT in some papers, is a separable fractional-order extension of the joint time-vertex Fourier transform for signals defined simultaneously on a graph and along a temporal axis. For a time-varying graph signal $\mathbf{X}\in\mathbb{C}^{N\times T}$, JFRFT applies a fractional transform in time and a fractional transform on the graph, thereby interpolating continuously between the signal domain and the joint spectral domain. Within graph signal processing, it has become a framework for joint spectral analysis, denoising, Wiener filtering, sampling, and trainable spatiotemporal representations, and it now sits at the intersection of classical FRFT theory, graph Fourier analysis, product-graph processing, and learnable spectral operators (Alikaşifoğlu et al., 2022, Zhang et al., 22 May 2025, Wang et al., 13 Oct 2025).

1. Formal definition and operator model

A joint time-vertex signal is typically represented as a matrix

$\mathbf{X}\in\mathbb{C}^{N\times T},$

whose rows index graph vertices and whose columns index time samples, with vectorized form

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$

The temporal factor is a discrete fractional Fourier transform (DFRFT) matrix, while the vertex factor is a graph fractional Fourier transform (GFRFT) matrix constructed from a graph Fourier basis derived from a chosen graph operator such as an adjacency matrix, Laplacian, row-normalized adjacency, symmetric normalized adjacency, or normalized Laplacian (Alikaşifoğlu et al., 2022, Yan et al., 11 Sep 2025).

Under one widely used convention, the transform is written

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$

and in vectorized form

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$

Other papers write the same construction with paper-specific order symbols and adjoint conventions, for example

$$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$

The invariant structural point across these formulations is separability: the joint transform is the Kronecker product of a temporal fractional operator and a graph fractional operator (Alikaşifoğlu et al., 2022, Zhang et al., 22 May 2025).

The temporal fractional component is inherited from FRFT theory. In discrete form, one paper writes

$$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$

with $F^{0}=I$, $F^{1}=F$, and $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$. The graph fractional component is defined spectrally from a graph transform operator, for example

$\mathbf{X}\in\mathbb{C}^{N\times T},$0

or, in adjacency-based notation,

$\mathbf{X}\in\mathbb{C}^{N\times T},$1

Accordingly, JFRFT generalizes the ordinary joint time-vertex Fourier transform (JFT), which is recovered at unit orders, and reduces to the identity at zero orders (Zhang et al., 22 May 2025, Wang et al., 13 Oct 2025).

A recurrent notational issue is that different papers assign the symbols $\mathbf{X}\in\mathbb{C}^{N\times T},$2 and $\mathbf{X}\in\mathbb{C}^{N\times T},$3 differently to the time and graph domains. This does not alter the operator class; it only changes which symbol labels the temporal or vertex fractional order.

2. Algebraic structure and fundamental properties

The core algebraic properties of JFRFT follow from the corresponding properties of the temporal DFRFT and the graph GFRFT. The defining results are index additivity, reversibility, reduction to identity, and reduction to the standard JFT. In operator form,

$\mathbf{X}\in\mathbb{C}^{N\times T},$4

and

$\mathbf{X}\in\mathbb{C}^{N\times T},$5

Consequently,

$\mathbf{X}\in\mathbb{C}^{N\times T},$6

These formulas are explicit in the early JFRT paper and reappear in later sampling, filtering, and hybrid-transform work (Alikaşifoğlu et al., 2022, Zhang et al., 22 May 2025, Wang et al., 13 Oct 2025).

When the temporal and graph factors are unitary, JFRFT is unitary: $\mathbf{X}\in\mathbb{C}^{N\times T},$7 which yields norm preservation and Parseval-type identities. This is standard for undirected graphs with orthonormal Laplacian eigenvectors, and it also holds in hyper-differential constructions where the temporal and graph generators are skew-Hermitian (Alikaşifoğlu et al., 2022, Yan et al., 29 Jul 2025). However, this property is conditional rather than universal. For non-normal or directed adjacency-based graph operators, unitary behavior may fail, and some later learning-oriented papers do not assume Parseval or energy conservation as part of the method definition (Wang et al., 13 Oct 2025, Yan et al., 11 Sep 2025).

Separability is central both theoretically and computationally. In one formulation,

$\mathbf{X}\in\mathbb{C}^{N\times T},$8

so time and graph fractionalization can be applied independently and in either order. For ring or circular graph topologies, where the graph Fourier transform reduces to a DFT, the joint transform reduces to a two-dimensional DFRT; at unit orders this becomes the two-dimensional DFT (Alikaşifoğlu et al., 2022).

The Hilbert-space generalization extends the same idea from discrete time to continuous domains. There the joint operator is written

$\mathbf{X}\in\mathbb{C}^{N\times T},$9

acting on $$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$0, with time-vertex signals obtained as the case $$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$1. The corresponding coefficient representation is

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$2

and the paper proves additivity, commutativity, invertibility, and unitarity in this setting (Zhang et al., 2024).

3. Filtering, Wiener formulations, and denoising

JFRFT entered the literature not only as a representation tool but also as a filtering domain. One early development was Tikhonov regularization in the JFRT domain. For noisy data $$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$3, the estimate is defined by

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$4

with closed-form solution

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$5

where the joint spectral filter is diagonal and has entries

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$6

This construction ties JFRFT directly to fractional joint Laplacians and quadratic regularization on the time and graph dimensions (Alikaşifoğlu et al., 2022).

A related line of work formulates optimal time-vertex filtering through Wiener–Hopf equations on product graphs and then extends the formulation to fractional domains. In that setting, a fractional joint filter is written as

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$7

and the optimal coefficients satisfy the fractional Wiener–Hopf equation

$$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$8

The paper’s central point is that optimal fractional orders can outperform both ordinary-domain time-vertex filtering and static graph-only fractional filtering (Ge et al., 2022).

Later work made JFRFT explicitly trainable. The “Trainable Joint Time-Vertex Fractional Fourier Transform” paper constructs the transform in hyper-differential form: $$\mathbf{x} \triangleq \mathrm{vec}(\mathbf{X})\in\mathbb{C}^{NT}.$$9

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$0

and then defines

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$1

Because

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$2

the transform orders become differentiable parameters that can be optimized jointly with a diagonal spectral filter (Yan et al., 29 Jul 2025).

The same model-driven idea appears in the neural-network-aided JFRFFNet framework. Its pipeline is explicitly

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$3

with three layers, Adam optimization, mean-square error loss, and trainable transform orders and filter coefficients. The intended advantage is operation with partial prior information rather than complete signal-and-noise statistics (Yan et al., 11 Sep 2025).

A further development connects JFRFT to two-dimensional graph bi-fractional transforms. In that work, the JFRFT analogue of Wiener-style diagonal filtering is

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$4

and the same paper also gives a differentiable formulation for joint optimization of orders and diagonal filters (Wang et al., 13 Oct 2025).

4. Sampling theory and localized reconstruction

A major 2025 development recast JFRFT as a sampling framework for jointly bandlimited time-vertex signals. Let $$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$5 and $$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$6 denote observed vertex and time index sets, with support projectors

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$7

Fractional spectral projectors are defined by

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$8

and jointly by

$$\mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{X};G) := \mathbf{F}_G^{\alpha_v}\,\mathbf{X}\,(\mathbf{F}^{\alpha_t})^{\mathrm T},$$9

A signal is jointly supported and jointly bandlimited when

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$0

This gives a fractional-domain version of classical support-bandwidth duality (Zhang et al., 22 May 2025).

The perfect localization theorem states that a signal is perfectly localized over $$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$1 and $$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$2 if and only if

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$3

equivalently

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$4

For reconstruction from samples $$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$5, the recovery operator is

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$6

and perfect recovery holds when

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$7

This is the exact JFRFT counterpart of subspace sampling and pseudo-inverse reconstruction (Zhang et al., 22 May 2025).

The same paper develops optimal sampling set design through several criteria: $$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$8

$$\mathbf{F}_J^{\alpha_t,\alpha_v} := \mathbf{F}^{\alpha_t}\otimes \mathbf{F}_G^{\alpha_v}, \qquad \mathrm{JFT}^{\alpha_t,\alpha_v}(\mathbf{x};G) = \mathbf{F}_J^{\alpha_t,\alpha_v}\mathbf{x}.$$9

together with MinPinv, MaxSig, and MaxVol formulations. These are implemented greedily by adding one sample at a time according to the marginal gain of the chosen objective (Zhang et al., 22 May 2025).

For large-scale problems, explicit eigendecompositions can be replaced by localized operators: $$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$0 When $$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$1, this coincides with $$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$2. Polynomial approximation with Chebyshev or Lanczos expansions yields locality and reduces the graph-domain and time-domain costs to $$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$3 and $$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$4, respectively (Zhang et al., 22 May 2025). The Hilbert-space HGFRFT paper gives an operator-theoretic analogue: if $$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$5, then perfect reconstruction holds when

$$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$6

with reconstruction

$$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$7

This places JFRFT sampling within a broader functional-analytic framework (Zhang et al., 2024).

5. Relation to JFT, 2D graph bi-fractional transforms, and dynamic variants

JFRFT is best understood as one member of a family of separable joint transforms. The main neighboring constructions differ in whether the temporal factor is treated as a classical DFRFT, as a graph FRFT on a path graph, or as a time-varying multi-parameter fractional operator.

Transform	Operator form	Distinguishing feature
JFT	$$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$8	Ordinary joint Fourier analysis
JFRFT	$$\hat{\mathbf{X}} = \mathbf{F}_{\mathcal G}^{\beta}\,\mathbf{X}\,(\mathbf{F}^{\alpha})^{\mathrm H}, \qquad \hat{\mathbf{x}} = \mathbf{J}^{\alpha,\beta}\mathbf{x}, \quad \mathbf{J}^{\alpha,\beta} = \mathbf{F}^{\alpha}\otimes \mathbf{F}_{\mathcal G}^{\beta}.$$9	Independent fractional orders in time and graph
2D-GBFRFT	$$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$0	Two graph-fractional factors on a Cartesian product
DMPJFRFT	$$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$1	Time-varying multiple fractional parameters

JFRFT reduces to the ordinary JFT at unit orders and to the identity at zero orders. By contrast, 2D-GBFRFT assigns independent fractional orders to the two factor graphs of a Cartesian product $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$2, which removes the requirement that one factor be a temporal path. A hybrid operator then interpolates between them: $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$3 The reduction laws are explicit: $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$4 This hybridization formalizes when a classical temporal FRFT basis should be preferred and when a path-graph fractional basis is more appropriate (Wang et al., 13 Oct 2025).

An even more flexible extension is the Dynamic Multiple-Parameter JFRFT (DMPJFRFT), which assigns a distinct graph-order vector to each time instant and a multi-parameter temporal order vector. In vectorized form,

$$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$5

The paper establishes identity, reversibility, linearity, and conditional additivity/commutativity, and it states that DMPJFRFT reduces to JFRFT when the graph-order columns are identical across time and the temporal orders match the standard single-parameter setting (Cui et al., 20 Nov 2025).

These relationships clarify a common misconception: JFRFT is not simply “the” fractional transform for dynamic graph data. It is the single-order separable joint transform. Later work broadens that model either by replacing the temporal factor with another graph factor, as in 2D-GBFRFT, or by allowing the orders themselves to vary over time and spectral components, as in DMPJFRFT.

6. Empirical behavior, implementation costs, limitations, and open directions

Empirical studies consistently report that fractional orders away from the ordinary Fourier setting can improve energy compaction, denoising, or recovery. In the original JFRT paper, denoising on Molene and NOAA data improved over the $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$6 baseline, with reported best parameters near $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$7 for Molene, $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$8 for yearly NOAA, and $$F^{\alpha}[m,n] = \sum_{k=0}^{T-1} u_k[m] e^{j(\pi/2)k\alpha} u_k[n],$$9 for monthly NOAA (Alikaşifoğlu et al., 2022). In the sampling paper, a sunshine-data experiment with $F^{0}=I$0 and $F^{0}=I$1 achieved best NMSE $F^{0}=I$2 by MaxSigMin at $F^{0}=I$3, while sea clutter experiments found an optimum at $F^{0}=I$4 with minimum NMSE $F^{0}=I$5 (Zhang et al., 22 May 2025).

The hybrid JFRFT/2D-GBFRFT study provides a sharper view of where pure JFRFT is strong and where hybridization helps. On PM-25, JFRFT outperformed pure 2D-GBFRFT, with MSE $F^{0}=I$6 versus $F^{0}=I$7 at $F^{0}=I$8, and the hybrid matched or slightly improved JFRFT by tuning $F^{0}=I$9 close to $F^{1}=F$0. On COVID, however, 2D-GBFRFT and the hybrid strongly outperformed JFRFT; at $F^{1}=F$1, JFRFT gave $F^{1}=F$2 MSE while 2D-GBFRFT and the hybrid gave $F^{1}=F$3. On REDSB dynamic image deblurring, JFRFT reported average $F^{1}=F$4, $F^{1}=F$5, $F^{1}=F$6, whereas the hybrid reported $F^{1}=F$7, $F^{1}=F$8, and $F^{1}=F$9 (Wang et al., 13 Oct 2025).

Learning-based variants also report strong gains. The trainable JFRFT paper states that learned JFRFT reduces runtime drastically relative to grid search: for SST at $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$0, JFRFT-search required $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$1s, $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$2s, and $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$3s, while JFRFT-learn required $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$4s, $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$5s, and $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$6s (Yan et al., 29 Jul 2025). JFRFFNet, which embeds JFRFT-domain Wiener filtering into a neural model, was reported as best on five datasets and second-best on three. Its output SNR included $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$7 dB on SST, $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$8 dB on PEMS08, $F^{\alpha}F^{\beta}=F^{\alpha+\beta}$9 dB on PEMS-BAY, and $\mathbf{X}\in\mathbb{C}^{N\times T},$00 dB on Quality, always exceeding the corresponding GFRFFNet result on those datasets (Yan et al., 11 Sep 2025). DMPJFRFTNet later extended this line by allowing time-varying orders, with reported SNR gains over JFRFT and other transform baselines on PEMSD7(M), PEMS08, Quality, SST, and video tasks; for example, on PEMSD7(M) with $\mathbf{X}\in\mathbb{C}^{N\times T},$01 and adjacency GSO, DMPJFRFT-I-I achieved $\mathbf{X}\in\mathbb{C}^{N\times T},$02 dB versus JFRFT $\mathbf{X}\in\mathbb{C}^{N\times T},$03 dB (Cui et al., 20 Nov 2025).

Implementation cost remains a persistent constraint. Constructing a graph fractional operator typically requires eigendecomposition or Jordan decomposition, often $\mathbf{X}\in\mathbb{C}^{N\times T},$04, while naive dense application of the full joint operator is expensive. Later papers repeatedly exploit separability to avoid forming the $\mathbf{X}\in\mathbb{C}^{N\times T},$05 Kronecker matrix explicitly; instead they apply left and right multiplications by $\mathbf{X}\in\mathbb{C}^{N\times T},$06 and $\mathbf{X}\in\mathbb{C}^{N\times T},$07 factors (Wang et al., 13 Oct 2025, Yan et al., 29 Jul 2025). Numerical conditioning is a recurring issue for adjacency-based or Jordan-based constructions, especially on large graphs, defective operators, or Vandermonde-type parameterizations (Alikaşifoğlu et al., 2022, Cui et al., 20 Nov 2025).

The main limitations stated across the literature are consistent. Performance depends on selecting suitable fractional orders and, in sampling problems, suitable bandwidths. A single global pair of orders may be suboptimal for highly dynamic or heterogeneous data. Unitarity and energy preservation may fail for non-normal graph operators. Explicit eigendecomposition remains costly on large graphs. Open problems identified in the sampling and transform papers include data-driven learning of $\mathbf{X}\in\mathbb{C}^{N\times T},$08, adaptive bandwidth selection, directed or time-varying graph extensions, theoretical guarantees under model mismatch, and fast approximation schemes with provable error bounds for very large-scale settings (Zhang et al., 22 May 2025, Alikaşifoğlu et al., 2022).

In that sense, JFRFT now occupies a well-defined place in graph signal processing: it is the canonical separable fractional generalization of JFT, with mature formulations for algebra, denoising, Wiener filtering, and sampling, but with ongoing development toward adaptive, partially supervised, and dynamically parameterized spatiotemporal spectral models.