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

Understand

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 $F (\| x \|) = \mathbb E_{\xi} \big[ f \big(|\langle x,\xi \rangle| \big) \big]$ 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.