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Related Articles from SNS
On the Role of the Double Fourier Sphere Method in Fast Algorithms on SO(3)
arXiv:2602.06677v3 Announce Type: replace Abstract: We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a Wigner transform - mapping harmonic to Fourier coefficients - followed by a Fourier transform. We revisit this formulation and interpret the Wigner transform as an explicit realization of the DFS method,...
Coordinate-invariant flux-surface Fourier analysis in tokamaks
arXiv:2606.02901v1 Announce Type: new Abstract: The Fourier spectra of resonant quantities in tokamaks depend on the choice of magnetic coordinates, and an area weighting of the Fourier integrand preserves the resonant coefficients on rational surfaces. That result constrains only the resonant interior; the coordinate dependence of the external Fourier spectrum, which determines the coupling to Resonant Magnetic Perturbation (RMP) coils and error-field penetration, was left untreated. This...
Rotation-Parameterized Graph Fractional Fourier Transform: Definition, Properties, and Optimal Filtering
Announce Type: replace-cross Abstract: Graph spectral representations are fundamental in graph signal processing, providing a rigorous frameworkforanalyzing graph-structured data. The graph fractional Fourier transform (GFRFT) extends the graph Fourier transform (GFT) through a fractional-order parameter, enabling flexible spectral analysis with mathematical consistency. The angular graph Fourier transform (AGFT) further introduces angular control by rotating GFT eigenvectors; however,...
PJ-RoPE: A Fourier-Jet-Affine Position Space for Relative Attention
Announce Type: new Abstract: We unify RoPE's Fourier phase, Jordan-RoPE's finite jets, and ALiBi's affine recency into a single learnable relative-position space, and study which regions of this space are selected by different tasks. PJ-RoPE is a Fourier-Jet-Affine formulation for relative attention, with an optional Poincare-type reading as the affine completion of a homogeneous Fourier-jet positional representation. Algebraically, the same primitives form a finite constant-coefficient...
Anti-Fourier heat flux does not certify the fourth-order closure state of a rarefied cavity
arXiv:2606.01480v1 Announce Type: new Abstract: Cold-to-hot heat transfer in rarefied cavities is usually treated as a signature of Fourier-law failure. Here it is used to ask whether a correct anti-Fourier heat-flux field certifies the flux-side fourth-order closure state. In a two-dimensional monatomic flow, the heat-flux hierarchy observes the divergence of the composite R26-level tensor \(A_{ij}=R^{\cl}_{ij}+\Delta\delta_{ij}/3\), not the tensorial fourth-order anisotropy...
Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
Announce Type: new Abstract: The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by...
Learning DNF through Generalized Fourier Representations
arXiv:2506.01075v2 Announce Type: replace Abstract: The Boolean Fourier representation has been widely used in learning theory, particularly for learning Disjunctive Normal Form (DNF) under uniform and product distributions. Extending these results to non-product distributions has remained a longstanding open problem. We address this challenge by introducing a generalized Fourier representation that enables learning under a broad class of non-product distributions.
A high-order Fourier Continuation (FC)-based spectral incompressible Smoothed Particle Hydrodynamics (ISPH) scheme for general boundary conditions in wall-bounded domains
arXiv:2606.06247v1 Announce Type: new Abstract: In this paper, a high-order Fourier Continuation (FC) algorithm is introduced into the spectral smoothed particle hydrodynamics (SPH) scheme to simulate the wall-bounded incompressible flows. This work aims to extend the spectral ISPH scheme towards the high-order simulation of flows with non-periodic wall boundary conditions. Herein, a polynomial-based Fourier continuation technique is applied to the velocity and pressure to make the domain...
Further evidence towards the Fourier Entropy-Influence conjecture
arXiv:2606.00246v1 Announce Type: cross Abstract: The Fourier Entropy-Influence (FEI) conjecture states that the Fourier entropy of Boolean functions is uniformly bounded by their total influence. It has been verified for canonical examples such as disjoint tribes and for some classes of Boolean functions such as symmetric functions and read-$k$ decision trees (with a constant that depends linearly on $k$). In this note we present new classes of Boolean functions that verify the FEI conjecture.
Fourier--Galerkin Methods for Subwavelength Resonances in two-dimensional Acoustic Metamaterials
arXiv:2605.23251v2 Announce Type: replace Abstract: We present a Fourier--Galerkin asymptotic framework for the analysis and computation of subwavelength resonances in two-dimensional scattering problems in finite domains. Starting from the boundary integral formulation, we apply a Fourier--Galerkin discretization to derive an explicit finite-dimensional effective matrix whose kernel characterizes the resonant frequencies. In the subwavelength regime, we obtain asymptotic expansions of this...