Doctor of Philosophy (PhD)
The cosine-λ transform, denoted Cλ, 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 Cλ extends meromorphically in λ to the complex plane with poles at (among other values) λ =-1,…, -p. In this dissertation we normalize Cλ 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.
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Cross, Christopher Adam, "Partial Cosine-Funk Transforms at Poles of the Cosine-λ Transform on Grassmann Manifolds" (2015). LSU Doctoral Dissertations. 748.