The Fourier Transform of the Stiefelian Surface Measure

TL;DR

This paper computes the leading term of the Fourier transform of the measure on the Stiefelian surface, revealing asymptotic behavior in most directions and connecting to various mathematical concepts.

Contribution

It provides the first asymptotic expansion of the Fourier transform of the Stiefelian surface measure for most directions, using stationary phase methods.

Findings

Derived the first term of the Fourier transform's asymptotic expansion.

Identified connections to trace moments, random walks, and Bessel functions.

Highlighted unknown behavior in certain directions.

Abstract

Let $\operatorname{St}^{n}_{k}\subset\mathbb{R}^{n\times k}$ be the set of all $n\times k$ matrices whose columns are mutually orthogonal and of unit Euclidean length, and let $\mu_{n,k}$ be the surface measure corresponding to this embedding. We calculate the first term of the asymptotic expansion of the Fourier transform of $\mu_{n,k}$ for most directions, using the method of stationary phase. The asymptotic behavior near the remaining directions is unknown. We note some interesting connections to trace moments of orthogonal matrices, discrete random walks, Bessel functions, and pose some questions.