Gaussian Analogue of Fourier Transform

Updated 13 October 2025

Gaussian analogues are generalized Fourier transforms that use Gaussian-type functions as eigenfunctions, ensuring key properties like invariance and convolution-multiplication duality.
The methodology encompasses q-Fourier transforms, dilation-invariant kernels, and discrete analogues, with inversion formulas extended to nonstandard settings.
Applications span signal processing, quantum physics, and statistical mechanics, leveraging these transforms for improved spectral resolution and time-frequency analysis.

The Gaussian analogue of the Fourier transform encompasses a diverse set of generalizations and reinterpretations of the classical Fourier transform, motivated by the ubiquity and special mathematical properties of the Gaussian function and its generalizations (q-Gaussians, Hermite-Gauss functions, etc.) in analysis, probability, statistical mechanics, harmonic analysis, signal processing, and quantum physics. These approaches preserve or adapt key features of the Fourier transform—such as invariance, convolution-multiplication duality, and eigenfunction structure—so that Gaussian-like functions remain central in the associated transform theory.

1. Definition and Foundations

The classical Fourier transform $\mathcal{F}$ acting on $f \in L^2(\mathbb{R})$ is characterized by its invariance properties and the fact that the Gaussian $g(x) = \exp(-x^2/2)$ is an eigenfunction: $\mathcal{F} g = g$ . More generally, a "Gaussian analogue" refers to integral transforms that (i) admit a Gaussian or its generalization as a (possibly distinguished) eigenfunction, (ii) intertwine with important operations such as dilations or translations, and (iii) reproduce many structural properties of the classical Fourier transform.

Generalizations include:

q-Fourier Transform: Defined as $F_q[f](\xi) = \int_{-\infty}^{\infty} f(x) e_q^{i\xi x [f(x)]^{q-1}} dx$ for $1 \leq q < 3$ , with the q-exponential $e_q^z = [1 + (1-q)z]^{1/(1-q)}$ (Jauregui et al., 2010).
Family of Dilation-Invariant Transforms: Characterized by composition with dilations and preserving $n$ -Gaussians $g_n(x) = \exp(-x^{2n}/2n)$ as eigenfunctions (Williams et al., 2014).
Transforms in Finite and Discrete Settings: Finite Gaussians and discrete analogues constructed via theta functions and Jacobi polynomials, with self-similar Fourier transformation behavior (Cotfas, 2015, Cotfas, 2019).
Transforms Built from Gaussian Process Regression: Bayesian analogues where the posterior mean in a GP model yields a Fourier transform analytically tractable via covariance kernel manipulation (Ambrogioni et al., 2017).

2. Generalized Transform Structures and Inversion Formulas

Nonlinear extensions (such as the q-Fourier transform) introduce significant structural modifications:

The standard inversion formula $\mathcal{F}^{-1}f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-i\xi x} \mathcal{F}f(\xi) d\xi$ does not directly generalize under non-additive exponents.
For the q-Fourier transform, inversion is achieved by integrating the q-transform over all translations:

$f(y) = \left[ \frac{2-q}{\gamma\pi} \int_{-\infty}^\infty F_q[f^{(y)}](\xi, y) d\xi \right]^{1/(2-q)}$

with $\gamma$ depending on the support of $f$ (Jauregui et al., 2010). This leverages the q-generalized Dirac delta:

$\delta_q(x) = \frac{2\pi}{2-q} \int_{-\infty}^{\infty} e_q^{i\xi x} d\xi,$

which, under suitable integrability conditions, acts analogously to the classical Dirac delta for function recovery.

Integrals involving Gaussian-modulated Bessel functions and discrete Gaussian sums underpin exact analytic calculations in physical applications (notably, acoustic radiation integrals and discrete oscillator models) (Carley, 2013, Cotfas, 2015).

3. Eigenfunctions, Kernel Construction, and Transform Families

A central organizing principle is the identification of eigenfunctions:

Dilation Eigenfunction Principle: The kernel of a generalized transform is constructed so that a Gaussian-like function is an eigenfunction, leading to unitarity ( $\|\mathcal{F}f\| = \|f\|$ ) and cyclic spectral properties (e.g., $\mathcal{F}^4 = {\rm Identity}$ ), preserved in broader families indexed by $n$ (Williams et al., 2014).
Hermite-Gauss and Harmonic Gaussian Functions: In time-frequency analysis, harmonic Gaussian functions modulated by Hermite polynomials provide localized bases, enabling time-frequency energy distributions and exact signal reconstruction formulas (Ranaivoson et al., 2013). The Fourier transforms of these functions reproduce the Gaussian window structure modulated appropriately in frequency domain.

Discrete kernels such as the Jacobi theta function and Gauss hypergeometric functions ${}_2F_1$ generalize the eigenfunction framework to finite/digital settings and lead to new reconstructive transforms and inversion formulas (Yakubovich, 2020, Cotfas, 2019).

4. Functional Analysis, Algebraic Structure, and Operator Theory

In infinite-dimensional settings such as Wiener space, Gaussian-based transforms are extended to analyze path functionals and Gaussian processes:

Fourier-Feynman Transforms: $L_2$ analytic Fourier-Feynman transforms are linear operator isomorphisms acting on spaces of cylinder functionals, preserving Hilbert space structure and admitting inversion via explicit relations $T_{q, h}^{(2)^{-1}} = T_{q, -h}^{(2)}$ (Chang et al., 2015).
Algebraic Approaches: The Gaussian subalgebra of the Schwartz class forms an integral domain closed under convolution, and its fraction field serves as an operational setting for invertible differential operators—illuminating connections with Mikusiński's operational calculus (Rosenkranz et al., 2020).

5. Applications in Signal Processing, Physics, and Mathematics

Practical applications are widespread:

Complex Systems and Nonextensive Statistical Mechanics: Many experimental distributions are q-Gaussians, necessitating q-Fourier inversion tools for probability recovery in correlated systems (Jauregui et al., 2010, Rodrigues et al., 2016).
Time-frequency Analysis: Harmonic Gaussian functions provide energy representations with positive definiteness and multi-scale adaptability well-suited for nonstationary signal decomposition (Ranaivoson et al., 2013).
Numerical Quantum Dynamics: Gaussian wave packet transforms, especially those discretized via Gauss–Hermite quadrature, afford efficient representations for wave functions and high accuracy in quantum molecular simulations (Bergold et al., 2020).
Spectral Estimation via Gaussian Process Regression: The Bayes-Gauss-Fourier transform yields sharply resolved spectra from finite data, with improved sidelobe and noise suppression over classical DFT (Ambrogioni et al., 2017).
Integral Geometry and Valuation Theory: The Alesker-Fourier transform translates classical Fourier operations into convex geometric valuations, intertwining product and convolution structures and providing inversion and adjointness results directly analogous to functional Fourier theory (Faifman et al., 2023).

6. Connections to Uncertainty Principles and Limit Laws

Gaussian analogues often serve as extremal functions for uncertainty principles:

Hardy-type Theorem for Deformed Transforms: Generalizations such as the $(k, \frac{2}{n})$ –Fourier transform admit Hardy’s theorem with Gaussian-type decay as the critical threshold; the Gaussian remains the unique extremiser in these uncertainty relations, and the same threshold demarcates the triviality and saturation cases—even under temporal evolution via associated heat equations (Jilani et al., 3 Mar 2025).
Central Limit Theorem in Correlated Systems: The q-Gaussian's role as an attractor for generalizations of the central limit theorem underscores the necessity for q-Fourier methods (Jauregui et al., 2010, Rodrigues et al., 2016).

7. Algorithmic, Computational, and Further Directions

Algorithmic implementations benefit from the structure of Gaussian analogues:

Weighted Sums and Complex Error Functions: Approximating the Fourier transform as a weighted sum over Gaussians, expressible through complex error functions, enables non-periodic wavelet-like representations with rapid convergence (Abrarov et al., 2015).
Reduction of Multidimensional Integrals: In quantum amplitude calculations, alternative transforms related to Gaussians simplify multidimensional reductions, handling singularities and integrals over special functions such as Macdonald and Meijer G-functions (Straton, 2022, Benhaddou, 2017).
Spectral Adaptivity: Families of transforms parametrized by the degree or exponent $n$ allow the kernel’s frequency localization to be tailored to specific signal attributes, offering improved filter design and time–frequency concentration (Williams et al., 2014).

Future research directions include understanding the boundedness and invariance properties of generalized kernels, extending the operator-theoretic framework to more general spaces, and exploring the deep connections between these Gaussian analogues, spectral theory, and modern probability.

In summary, the Gaussian analogue of the Fourier transform unifies a multitude of generalizations and reinterpretations, preserving the centrality of Gaussian-type functions in harmonic analysis, operator theory, time-frequency methods, and physical applications. These frameworks maintain the essential algebraic, analytic, and computational features of the classical Fourier transform, extending its reach to nonextensive, nonlinear, discrete, and operator-theoretic settings where Gaussian-like functions continue to play foundational roles.