Explicit computations of Fourier transforms of polyhedral cones

TL;DR

This paper introduces a new polynomial interpolation method for computing Fourier transforms of polyhedral cones, enabling efficient calculations under certain combinatorial conditions despite the general #P-hardness of related volume computations.

Contribution

The paper presents a novel polynomial interpolation approach for Fourier transforms of cones, improving efficiency with specific combinatorial assumptions.

Findings

Efficient computation of Fourier transforms for cones with certain combinatorial structures.

Polynomial interpolation method applied to Fourier transforms of polyhedral cones.

Demonstrates limitations due to #P-hardness in the general case.

Abstract

The Fourier transforms of polyhedral cones can be used, via Brion's theorem, to compute various geometric quantities of polytopes, such as volumes, moments, and lattice-point counts. We present a novel method of computing these conic Fourier transforms by polynomial interpolation. Given the fact that computing volumes of polytopes is #P-hard (Dyer--Frieze [DF88]), we cannot hope for fast algorithms in the general case. However, with extra assumptions on the combinatorics of the cone, we demonstrate it is possible to compute its Fourier transform efficiently.