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Slicing of Radial Functions: a Dimension Walk in the Fourier Space

Nicolaj Rux, Michael Quellmalz, Gabriele Steidl

TL;DR

The paper addresses recovering a univariate radial slice $f$ from a radial high-dimensional function $F$ linked by $F(|x|)=\mathbb{E}_{\xi}[f(|\langle x,\xi\rangle|)]$. It develops a Fourier-space dimension-walk using the rotation $\mathcal{R}_d$, averaging $\mathcal{A}_d$, and multiplication $\mathcal{M}_d$ operators to reduce from $d$ to 1D, and extends the theory from $L^1$ radial functions to tempered distributions and positive definite kernels via Bochner’s theorem. The main contributions include explicit inversion formulas (e.g., $f=\tfrac{\omega_{d-1}}{2}(\mathcal{F}_1\circ\mathcal{M}_d)[\rho]$ with $\rho$ tied to $F$), a distributional slicing framework for radial tempered distributions, and a positive definite correspondence between radial $F$ and $f$. These results enable rigorous, efficient handling of high-dimensional radial convolutions in applications while providing theoretical guarantees on smoothness and positivity properties. The work integrates Abel-type relations, Radon-adjoint connections, and fractional derivative perspectives to unify both function and distributional settings.

Abstract

Computations in high-dimensional spaces can often be realized only approximately, using a certain number of projections onto lower dimensional subspaces or sampling from distributions. In this paper, we are interested in pairs of real-valued functions $(F,f)$ on $[0,\infty)$ that are related by the projection/slicing formula $F (\| x \|) = \mathbb E_ξ \big[ f \big(|\langle x,ξ\rangle| \big) \big]$ for $x\in\mathbb R^d$, where the expectation value is taken over uniformly distributed directions in $\mathbb R^d$. While it is known that $F$ can be obtained from $f$ by an Abel-like integral formula, we construct conversely $f$ from given $F$ using their Fourier transforms. First, we consider the relation between $F$ and $f$ for radial functions $F(\| \cdot\| )$ that are Fourier transforms of $L^1$ functions. Besides $d$- and one-dimensional Fourier transforms, it relies on a rotation operator, an averaging operator and a multiplication operator to manage the walk from $d$ to one dimension in the Fourier space. Then, we generalize the results to tempered distributions, where we are mainly interested in radial regular tempered distributions. Based on Bochner's theorem, this includes positive definite functions $F(\| \cdot\| )$ and, by the theory of fractional derivatives, also functions $F$ whose derivative of order $\lfloor d/2\rfloor$ is slowly increasing and continuous.

Slicing of Radial Functions: a Dimension Walk in the Fourier Space

TL;DR

The paper addresses recovering a univariate radial slice from a radial high-dimensional function linked by . It develops a Fourier-space dimension-walk using the rotation , averaging , and multiplication operators to reduce from to 1D, and extends the theory from radial functions to tempered distributions and positive definite kernels via Bochner’s theorem. The main contributions include explicit inversion formulas (e.g., with tied to ), a distributional slicing framework for radial tempered distributions, and a positive definite correspondence between radial and . These results enable rigorous, efficient handling of high-dimensional radial convolutions in applications while providing theoretical guarantees on smoothness and positivity properties. The work integrates Abel-type relations, Radon-adjoint connections, and fractional derivative perspectives to unify both function and distributional settings.

Abstract

Computations in high-dimensional spaces can often be realized only approximately, using a certain number of projections onto lower dimensional subspaces or sampling from distributions. In this paper, we are interested in pairs of real-valued functions on that are related by the projection/slicing formula for , where the expectation value is taken over uniformly distributed directions in . While it is known that can be obtained from by an Abel-like integral formula, we construct conversely from given using their Fourier transforms. First, we consider the relation between and for radial functions that are Fourier transforms of functions. Besides - and one-dimensional Fourier transforms, it relies on a rotation operator, an averaging operator and a multiplication operator to manage the walk from to one dimension in the Fourier space. Then, we generalize the results to tempered distributions, where we are mainly interested in radial regular tempered distributions. Based on Bochner's theorem, this includes positive definite functions and, by the theory of fractional derivatives, also functions whose derivative of order is slowly increasing and continuous.
Paper Structure (14 sections, 22 theorems, 175 equations, 2 figures)

This paper contains 14 sections, 22 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rotating, Averaging and Slicing
  3. Slicing of $L^1$ Functions
  4. Slicing of Tempered Distributions
  5. Radial Schwartz Functions
  6. Radial Tempered Distributions
  7. Slicing of Radial Regular Tempered Distributions
  8. Slicing of Positive Definite Functions
  9. Conclusions
  10. Proof of Theorem \ref{['thm:slicing_b']}
  11. Proof of Theorem \ref{['thm:slicing']}
  12. Proof of Theorem \ref{['prop def of avg']}
  13. Proof of Theorem \ref{['thm:dist slicing']}
  14. Riemann--Liouville Fractional Integrals and Derivatives

Key Result

Theorem 2.1

Let $d \in \mathbb N$, $d \ge 2$ and let $f \in L^1_\mathrm{loc}([0,\infty))$. Then the function $F\colon [0,\infty) \to \mathbb R$ fulfilling the slicing relation is determined by the Abel-type integral where $c_d \coloneqq \frac{2 \omega_{d-2}}{\omega_{d-1}}$ and $\omega_{d-1}= \frac{2\pi^{d/2}}{\Gamma(d/2)}$ denotes the surface measure of $\mathbb S^{d-1}$.

Figures (2)

  • Figure 1: Projection of points $x_1,\dots,x_5\in\mathbb{R}^2$ onto the line in direction $\xi$.
  • Figure 2: Abel-type transform of the confluent hypergeometric distribution $f$ results in the Gaussian $F$ ($d=10$).

Theorems & Definitions (43)

  • Theorem 2.1
  • Remark 2.2: Connection with the Radon transform
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Corollary 3.4
  • proof
  • Example 3.5
  • ...and 33 more