Stationary Transform Vector

• Stationary transform vector is a concept defining invariant properties under transformations like Fourier, Laplace, and group actions in stationary systems.
• It provides critical insights into gradient orientations in 2D distance transforms, invariant trajectories in Minkowski spacetime, and fixed-point behavior in higher-order Markov tensors.
• Its applications span adaptive signal coding, Bayesian time series, and even quantum field analysis around black holes, illustrating its interdisciplinary impact.

A stationary transform vector is a concept appearing across multiple branches of mathematical physics, signal processing, and stochastic analysis, typically denoting an invariant vector (or field) under a specified transformation in a stationary system. In its most fundamental instantiations, a stationary transform vector characterizes the structural properties, invariant features, or quantum numbers associated with stationary or steady-state phenomena within the framework of transformations (Fourier, Laplace, Mellin, coordinate transformations, or group actions).

1. Stationary Transform Vectors via Complex Wave Representation and Stationary Phase

In the context of gradient density estimation on 2D distance transforms (Gurumoorthy et al., 2011), the stationary transform vector emerges from the analysis of the complex wave representation (CWR) φ(X) = exp(iS(X)/τ), where S(X) is the unsigned Euclidean distance transform and τ a vanishing free parameter. The Fourier transform of φ, for τ → 0, is sharply concentrated on the unit circle in frequency space due to the stationary phase approximation. Here, the transform vector refers to the spatial frequency or orientation bin corresponding to the local gradient ∇S (with ‖∇S‖ = 1 a.e.), i.e., the normalized power spectrum converges to the gradient orientation density function,

$$P(\theta) = \frac{1}{L} \sum_k \frac{R_k^2(\theta)}{2}$$

where R_k(θ) is the Voronoi ray length associated with source point Y_k in the gradient direction θ. Thus, stationary transform vectors provide frequency space localization reflecting spatial gradient orientations, underpinning histogram-of-gradient constructions and Fourier-based density estimations without explicit differentiation.

2. Invariant Vectors in Group-Theoretic and Geometric Contexts

In Lorentzian geometry (Bunney, 2023), stationary trajectories in Minkowski spacetime are classified by the conjugacy classes of the restricted Poincaré group ISO⁺(n,1); each class corresponds to a set of timelike Killing vectors,

$\xi^\mu = c^\mu + \omega^{\mu\nu} x_\nu$

where c^μ denotes translation, ω^{μν} is an antisymmetric matrix encoding boosts and rotations, and x_ν are coordinates. The stationary transform vector here is the generator (up to conjugacy) of a stationary trajectory—the integral curve of a timelike Killing vector—invariant under a specified isometry. This leads, via the generalized Frenet–Serret formalism, to families of stationary worldlines parametrized by constant curvature coefficients, with explicit characterization in higher-dimensional Minkowski spaces.

3. Stationary Transform Vectors in Markov Tensors and Fixed-Point Problems

For higher-order transition probability tensors (Huang et al., 2018), a stationary probability vector is a fixed point x = (x₁, x₂, …, xₙ)ᵗ satisfying

$$\sum_{i_2,…,i_m=1}^2 P_{i, i_2,…,i_m} x_{i_2}…x_{i_m} = x_i, \quad \sum_{i=1}^n x_i = 1$$

and for two-dimensional symmetric tensors, generically (½,½)ᵗ is the unique stationary vector, barring special reducible tensor structures. In this setting, a stationary transform vector identifies the unique invariant distribution under the tensor-induced transformation, reflecting convergence and ergodic properties in higher-order Markov chains.

4. Multidimensional Laplace Transform: Functional Analytic Perspective

A substantial generalization of stationary transform vectors arises in the theory of multidimensional vector-valued Laplace transforms (Kostic, 29 Apr 2025), where the transform

$$\mathcal{L}\{f\}(x_1, …, x_n) = \int_0^\infty … \int_0^\infty e^{-(x_1 t_1 + … + x_n t_n)} f(t_1, …, t_n) dt_1…dt_n$$

is defined for functions f taking values in a sequentially complete locally convex space. Here, stationary transform vectors denote elements in the transform domain—typically solutions to stationary algebraic equations derived from transformed Volterra-type integro-differential inclusions:

$$B u(t_1,…,t_n) \in A \int … a(\cdot) u(\cdot) + C f(\cdot)$$

whose Laplace-transformed analog is an equation for $\mathcal{L}\{u\}(x_1,…,x_n)$, i.e., a time-invariant "stationary" vector in the image space. Operational properties such as translation invariance, commutation with continuous linear operators, inversion via multidimensional Post-Widder formulas, and region-of-convergence characterizations undergird these applications.

5. Transform Vectors in Stochastic Integration and Spectral Analysis

In stochastic analysis (Alpay et al., 2011), the extension of the S-transform to m-noise spaces permits pathwise definition of stochastic integrals (Wick–Itô type) for Gaussian stationary increment processes. The stationary transform vector analog is encoded in the spectral density parameterization and the transformed expectation functional,

$$(S_m\Phi)(s) = E[:\exp(\langle w,s\rangle): \Phi(w)]$$

as well as in the associated Itô-type formula where the stationary increment property is crucial. The process' transform vector is then linked to spectral representations of increments and the operational calculus in white noise analysis.

6. Stationary Transform Vectors in Adaptive Transform Coding and Statistics

In adaptive signal coding (Boragolla et al., 2022), a stationary transform vector refers to the orthonormal matrix (codeword) optimally selected from a codebook to minimize distortion for a stationary block of a nonstationary vector process. The algorithm operates on the Stiefel manifold to ensure matrix orthonormality, alternating between partitioning covariance matrices and optimizing transform vectors via steepest descent. In Bayesian inference on stationary vector autoregressions (Binks et al., 2023), the role of stationary transform vectors is fulfilled by the partial autocorrelation matrices (and their unconstrained transforms $A$), which encode model order and stationarity in time series, permitting Hamiltonian Monte Carlo sampling over unconstrained parameter spaces.

7. Physical Applications: Black Hole Physics and Massive Field Multipoles

In general relativity and field theory (Santos et al., 2020, Wu, 2022), stationary transform vectors are manifest in stationary scalar and vector clouds around Kerr–Newman black holes, emerging from synchronisation conditions (ω/m_j = Ω_H) and parametrized by quantum numbers (n, ℓ, j, m_j) that index stationary superradiant bound states. Analogously, multipole expansions for stationary massive vector and tensor fields introduce stationary transform vectors as irreducible Cartesian moment tensors, with physical implications for mass, spin, and gauge properties, and yielding modifications such as Yukawa-type radial modulations.

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The stationary transform vector framework thus unites fixed points of functional, geometric, or operator-induced transformations in stationary schemes—whether as spectral densities, group-theoretic generators, optimal codebook elements, or quantum numbers. Its formalization facilitates analysis and computation in invariant regimes of diverse physical, statistical, and mathematical systems.