Fourier Representation of Kernels

Updated 4 September 2025

Fourier representation of kernels is a method that expresses kernel functions via spectral expansions, linking their structure to fundamental properties like positive definiteness.
It enables efficient computational strategies, such as random Fourier features and sparse index sets, to improve kernel approximations in high-dimensional spaces.
The approach reveals insights into non-uniqueness, structural invariants, and applications ranging from harmonic analysis to algebraic geometry.

The Fourier representation of kernels refers to expressing, analyzing, or constructing kernels (integral operators, covariance functions, or interaction laws) through their Fourier, generalized Fourier, or spectral expansions. This perspective permeates diverse areas such as harmonic analysis, statistical learning theory, computational mathematics, algebraic geometry, signal processing, and numerical analysis. The Fourier-analytic viewpoint provides insights into positive definiteness, kernel structure, computational strategies, approximation quality, and non-uniqueness phenomena.

1. Fundamental Principles of Fourier Representation of Kernels

A kernel is a bivariate function $K(x, y)$ often assumed to be symmetric, positive definite, and sufficiently regular. The Fourier representation seeks to encode $K$ in terms of oscillatory bases, spectral densities, or generalized Fourier series. In the translation-invariant case on $\mathbb{R}^d$ , Bochner's theorem states that a continuous, shift-invariant, positive definite kernel can be written as:

$K(x, y) = \Phi(x-y) = \int_{\mathbb{R}^d} e^{i\omega \cdot (x - y)}\, d\mu(\omega),$

where $d\mu$ is a finite nonnegative measure (the spectral measure of $K$ ). This spectral viewpoint generalizes to operator-valued kernels (Brault et al., 2016, Minh, 2016), harmonizable (non-stationary) kernels (Shen et al., 2018), and even to non-Euclidean domains such as products of circles (Guella et al., 2015) or complex variables (Chakrabarti et al., 2018, Colombo et al., 2021).

For translation-invariant kernels on $\mathbb{R}$ :

The Fourier transform of the kernel $\hat{\Phi}(\omega)$ completely characterizes the kernel.
One can construct orthonormal expansions of the kernel's reproducing kernel Hilbert space (RKHS) using Fourier-analytic techniques (Tronarp et al., 2022).

The Fourier representation enables explicit construction of orthonormal bases, efficient numerical schemes, and unpacks structural properties such as positive definiteness and interpolation power.

2. Advanced Methodologies for Kernel Construction and Expansion

Fourier representation methodologies fall into several categories:

Method	Key Formula/Construction	Typical Application Scope
Spectral/Basis Expansion	$K(x, y) = \sum_k \hat{K}_k e^{2\pi i k \cdot (x-y)}$	Periodic kernels, interpolation, RKHS
Generalized Fourier Series	$W'(x) = \sum_{k=0}^K \beta_k \psi_k(x)$ , with $\{\psi_k\}$ orthogonal in $L^2(\rho)$	Learning interaction kernels (Pavliotis et al., 8 May 2025)
Operator-valued feature maps	$K(x, t) = \int e^{-i\langle \omega, x-t \rangle } A(\omega) d\mu(\omega)$ , $A(\omega)$ psd	Multi-task/multi-output kernels
Fourier (Chebyshev) modes on Spheres	$K((x, z), (y, w)) = \sum_{k,l} a_{k,l} P_k(x\cdot y) P_l(z\cdot w)$	Products of spheres/circles (Guella et al., 2015)
Fractional Integral Transform (Slicing)	$F(s) = C \int_0^1 f(ts) (1-t^2)^{\alpha} dt$	Dimension reduction (Hertrich, 16 Jan 2024)

For Mercer kernels, the classical integral operator eigen-decomposition can be bypassed using Fourier-analytic methods, resulting in closed-form analytic expansions in terms of familiar function bases (Hermite, Laguerre) (Tronarp et al., 2022). In more geometric or high-dimensional problems, slicing and reduction strategies using fractional integrals are employed to approximate summations efficiently (Hertrich, 16 Jan 2024).

In algebraic geometry, the concept of a "kernel" connects to derived category functors via Fourier–Mukai theory, where functors are defined by integral transforms with "kernels" in the derived category $D^b(X_1 \times X_2)$ (Canonaco et al., 2010, Rizzardo, 2012).

3. Existence, Uniqueness, and Structural Properties

The Fourier representation often reveals primary structural attributes:

Positive Definite Kernels: Bochner's theorem for translation-invariant kernels and its generalizations to operator-valued or non-stationary covariances (Csordas, 2013, Brault et al., 2016, Minh, 2016, Shen et al., 2018) hinge upon the non-negativity of the spectral measure or positivity/concavity properties of associated functions.
Non-Uniqueness of Kernel Representations: For integral transforms between derived categories (Fourier–Mukai functors), kernel objects need not be unique. Two distinct kernels $E_1, E_2 \in D^b(X \times X)$ may yield the same functor, and the assignment from kernel to functor is generally neither injective nor full (Canonaco et al., 2010). However, the induced cohomology sheaves are uniquely determined, so derived, K-theoretic, or cohomological invariants are canonical even though the kernels themselves are not (Canonaco et al., 2010).

Aspect	Scalar/Fourier Case	Fourier–Mukai (Algebraic Geometry)
Uniqueness	Spectral density uniquely determined	Kernel not unique, functor is
Faithfulness/Fullness	Yes (if positive definite)	No (not full nor essentially injective)
Cohomological invariants	Uniquely determined	Sheaf and K-theory classes unique

Strict Positive Definiteness in Harmonic Analysis: The support set of Fourier/Chebyshev coefficients characterizes strict positive definiteness for kernels on products of circles, requiring that every translate of every rectangular lattice in $\mathbb{Z}^2$ is hit by the support (Guella et al., 2015).

4. Computational Strategies and Approximation Methods

The Fourier framework enables efficient numerical approximation and learning:

Random Fourier Features (RFF): For large-scale machine learning, kernels are approximated via random sampling in the Fourier domain, replacing $K(x, y)$ by Monte Carlo averages over sampled frequencies (Brault et al., 2016, Minh, 2016). Extensions to operator-valued kernels, in multi-output or multi-task settings, use matrix-valued Fourier "weights" and concentration inequalities for error guarantees.
Index Set Fourier Series Features (ISFSF): For high-dimensional periodic kernels, deterministic sparse index sets provide more accurate expansions than random feature approaches, yielding substantially lower prediction error and better generalization with fewer features (Tompkins et al., 2018).
Slicing and FFT-based Summation: For radial kernels on $\mathbb{R}^d$ , radial integrals are reduced to one-dimensional problems using generalized Riemann–Liouville fractional integrals, followed by fast Fourier summation. Dimension-independent error bounds are established, notably for the Gaussian kernel (Hertrich, 16 Jan 2024).
Sparse Fourier Domain Learning: Parameter-efficient, scalable methods for learning continuous convolutional kernels via sparse sampling and updating in the Fourier domain address both computational and spectral bias constraints, effectively allowing large kernels without an explosion in parameters (Harper et al., 15 Sep 2024).
Diffusion Maps with Asymmetric Kernels: Expanding kernels directly in a tensor product Fourier basis enables efficient use of 2-D FFTs to facilitate dimensionality reduction and diffusion-distance computation for asymmetric kernels, dramatically improving on the computational cost compared to classical (SVD-based) methods (Gomez et al., 20 Jan 2024).

5. Theoretical Ramifications and Applications

The Fourier representation of kernels has several significant implications:

Eigenfunction Expansions in RKHS: Explicit orthonormal bases constructed via Fourier convolution yield tractable decompositions for Matérn, Gaussian, and Cauchy kernels, facilitating both theoretical investigation and low-rank approximations in numerical algorithms (Tronarp et al., 2022).
Higher Order Kernel Construction: Fourier transformations of suitably regular covariance functions with prescribed vanishing derivatives at the origin produce kernels of arbitrarily high order, supporting kernel density estimators with tunable properties and explicit mean integrated square error (MISE) control (Das et al., 2020).
Analytic Control in Harmonic Analysis and Number Theory: For entire functions represented as Fourier transforms of admissible kernels, properties such as positivity, log-concavity, and root distribution are connected to fundamental inequalities (e.g., generalized Laguerre inequalities) and conjectures like the Riemann Hypothesis (Csordas, 2013).
Algebraic Geometry and Derived Categories: The Fourier–Mukai formalism, while powerful, is not canonical at the object level. Nevertheless, its cohomological and categorical consequences—such as derived equivalence, functor cohomological footprints, and moduli theory—are determined by the Fourier representation up to invariants (Canonaco et al., 2010, Rizzardo, 2012).
Signal Processing and Clifford Analysis: Extensions of the Fourier framework to Clifford-valued functions and slice monogenic kernels lead to explicit Bessel- and Poisson-based kernel representations, spectral decompositions, and integral formulas for higher spin or fractional differential operators (Constales et al., 2015, Colombo et al., 2021).
Wave Front Sensing and Optical Systems: Modeling of integral operators in wave front sensing uses the Fourier kernel formalism to derive convolution models, impulse responses, and transfer functions, with particular emphasis on the interplay between physical domain parameters and Fourier characteristics (Fauvarque et al., 2019).

6. Limitations, Non-Uniqueness, and Open Problems

Fundamental limitations associated with the Fourier representation of kernels include:

Non-Uniqueness of Representation: In certain geometric or categorical settings, multiple kernels correspond to the same functor or operator; only derived or cohomological invariants are uniquely specified (Canonaco et al., 2010).
Support Conditions and Degeneracies: For positive (or strict positive) definiteness, the support of the Fourier expansion coefficients must intersect every relevant lattice or subgroup; missing harmonics can lead to degeneracy and ill-posedness (Guella et al., 2015).
Open Problems: Several open questions relate to the complete characterization of kernel properties (e.g., which log-concave admissible kernels yield positive definite associated transforms (Csordas, 2013)), peculiarities of high-frequency expansion (Gibbs phenomenon, spectral bias (Harper et al., 15 Sep 2024)), and computational issues in operator-valued or high-dimensional settings.
Spectral Bias and High-Frequency Learning: Neural approaches to learning kernel representations in the Fourier domain can be hindered by an intrinsic bias towards low-frequency features; strategies leveraging the Gibbs phenomenon in the Fourier domain can partially alleviate this, crucially for applications requiring high-frequency sensitivity (Harper et al., 15 Sep 2024).

7. Extensions to Non-Stationary, Asymmetric, and High-Dimensional Settings

Recent work generalizes the Fourier representation:

Harmonizable Kernels admit generalized spectral representations involving bivariate measures on frequency space, thus encompassing both stationary and broad classes of non-stationary kernels (Shen et al., 2018).
Asymmetric Kernels: The Fourier basis, via tensor products, enables explicit coordinate representations and fast computation of diffusion metrics in non-self-adjoint settings (Gomez et al., 20 Jan 2024).
High Dimensions: Slicing techniques and fractional integral transforms demonstrate that efficient and accurate kernel summations in high-dimensional spaces are possible using one-dimensional Fourier-based reductions, with sharp, dimension-independent error control (Hertrich, 16 Jan 2024).

Conclusion

The Fourier representation of kernels provides a unified language connecting harmonic analysis, numerical algorithms, algebraic geometry, learning theory, and stochastic modeling. It underpins efficient computational methods, facilitates theoretical insights (positive definiteness, spectral properties, uniqueness issues), and enables applications across scientific domains. The approach has proven especially fruitful in settings where structure, efficiency, or explicitness is paramount, while also revealing important subtleties regarding kernel non-uniqueness, support, degeneracies, and approximation error. Across these applications, the fusion of Fourier methods and kernel theory introduces powerful tools for both theory and computational practice.