Partial Cosine-Funk Transforms at Poles of the $\textrm{Cos}^\lambda$ Transform on Grassmann Manifolds

TL;DR

This paper studies the cosine-$ ext{lambda}$ transform on Grassmann manifolds, normalizes it, and evaluates it at poles to define partial cosine-Funk transforms, including an analog of the Funk transform.

Contribution

It introduces a normalization and pole evaluation of the cosine-$ ext{lambda}$ transform on Grassmannians, leading to new partial cosine-Funk transforms including a Funk-analog.

Findings

Defined partial cosine-Funk transforms at poles of the cosine-$ ext{lambda}$ transform.

Identified the transform at $ ext{lambda}=-p$ as an analog of the Funk transform.

Extended the theory of cosine transforms to Grassmann manifolds at special parameter values.

Abstract

The cosine-$\lambda$ transform, denoted $\mathcal{C}^\lambda$, is a family of integral transforms we can define on the sphere and on the Grassmann manifolds $\textrm{Gr}(p, \mathbb{K}^n) = \textrm{SU}(n,\mathbb{K})/\text{S}(\textrm{U}(p,\mathbb{K}) \times \textrm{U}(n-p,\mathbb{K}))$ where $\mathbb{K}$ is $\mathbb{R}$, $\mathbb{C}$ or the skew field $\mathbb{H}$ of quaternions. The family $\mathcal{C}^\lambda$ extends meromorphically in $\lambda$ to the complex plane with poles at (among other values) $\lambda =-1,\ldots, -p$. In this paper we normalize $\mathcal{C}^\lambda$ and evaluate at those poles. The result is a series of integral transforms on the Grassmannians that we can view as partial cosine-Funk transforms. The transform that arises at $\lambda = -p$ is the natural analog of the Funk transform in this setting.