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.
","abstract_html":"<p>The cosine-λ transform, denoted C<sup>λ</sup>, is a family of integral transforms we can define on the sphere and on the Grassmannian manifolds of p-dimensional subspaces in K<sup>n</sup> 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. </p> <p> The family C<sup>λ</sup> extends meromorphically in λ to the complex plane with poles at (among other values) λ =-1,…, -p. In this dissertation we normalize C<sup>λ</sup> 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 <em>partial</em> cosine-Funk transforms. The transform that arises at λ = -p is the natural Funk transform for the Grassmannians, which was introduced by B. Rubin.</p>","abstract_has_math":false,"creators":["Cross, Christopher Adam"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-01-01T08:00:00Z","date_published":"2015-01-01T08:00:00Z","updated_at":"2026-07-24T02:58:17Z","subjects":["integral transforms","harmonic analysis","Cλ transform"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07072015-194931","https://repository.lsu.edu/gradschool_dissertations/748"],"render_values":[{"text":"etd-07072015-194931","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/748","href":"https://repository.lsu.edu/gradschool_dissertations/748","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.748","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Cross, Christopher Adam"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-06-26"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:09:51Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["integral transforms","harmonic analysis","Cλ transform"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-07072015-194931","10.31390/gradschool_dissertations.748","https://repository.lsu.edu/gradschool_dissertations/748"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Partial Cosine-Funk Transforms at Poles of the Cosine-λ Transform on Grassmann Manifolds"]}]}],"canonical_facts":{"dc:creator":["Cross, Christopher Adam"],"dc:date":["2015-06-26"],"dc:date.available":["2022-05-12T23:09:51Z"],"dc:description.abstract":["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.
"],"dc:identifier":["etd-07072015-194931","10.31390/gradschool_dissertations.748","https://repository.lsu.edu/gradschool_dissertations/748"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["integral transforms","harmonic analysis","Cλ transform"],"dc:title":["Partial Cosine-Funk Transforms at Poles of the Cosine-λ Transform on Grassmann Manifolds"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:58:17Z"}