#### Identifier

etd-07072015-194931

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

The cosine-λ transform, denoted C^{λ}, is a family of integral transforms we can define on the sphere and on the Grassmannian manifolds of p-dimensional subspaces in K^{n} where K is R, C or the skew field H of quaternions. We treat the Grassmannians as the symmetric spaces SO(n)/S(O(p) × O(q)), SU(n)/S(U(p) × U(q)) and Sp(n)/(Sp(p) × Sp(q)) and we work by analogy with the case of the cosine-λ transform on the sphere, which is also a symmetric space.

The family C^{λ} extends meromorphically in λ to the complex plane with poles at (among other values) λ =-1,…, -p. In this dissertation we normalize C^{λ} and we use well known harmonic analysis tools to 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 λ = -p is the natural Funk transform for the Grassmannians, which was introduced by B. Rubin.

#### Date

2015

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Cross, Christopher Adam, "Partial Cosine-Funk Transforms at Poles of the Cosine-λ Transform on Grassmann Manifolds" (2015). *LSU Doctoral Dissertations*. 748.

https://digitalcommons.lsu.edu/gradschool_dissertations/748

#### Committee Chair

Olafsson, Gestur