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

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 $$\text{Gr}$(p, $\mathbb{K}$^n) = $\text{SU}$(n,$\mathbb{K}$)/$\text{S}$($\text{U}$(p,$\mathbb{K}$) \times $\text{U}$(n-p,$\mathbb{K}$))$where$$\mathbb{K}$$is$$\mathbb{R}$$,$$\mathbb{C}$$ortheskewfield$$\mathbb{H}$$ofquaternions\mathrm{.}Thefamily$$\mathcal{C}$^\lambda$extendsmeromorphicallyin$\lambda$ to the complex plane with poles at (among other values) $\lambda =-1,\dots ,-p$. 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.