Partial Cosine-Funk Transforms at Poles of the Cosine-λ Transform on Grassmann Manifolds

Identifier

etd-07072015-194931

Author

Christopher Adam Cross, Louisiana State University and Agricultural and Mechanical CollegeFollow

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 Kⁿ 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://repository.lsu.edu/gradschool_dissertations/748

Committee Chair

Olafsson, Gestur

DOI

10.31390/gradschool_dissertations.748