TY - JOUR
T1 - Quasianalyticity, uncertainty, and integral transforms on higher grassmannians
AU - Faifman, Dmitry
N1 - Publisher Copyright:
© 2023 Elsevier Inc.
PY - 2023/12/15
Y1 - 2023/12/15
N2 - We investigate the support of a distribution f on the real grassmannian Grk(Rn) whose spectrum, namely its nontrivial O(n)-components, is restricted to a subset Λ of all O(n)-types. We prove that unless Λ is co-sparse, f cannot be supported at a point. We utilize this uncertainty principle to prove that if 2≤k≤n−2, then the cosine transform of a distribution on the grassmannian cannot be supported inside any single open Schubert cell Σk. The same holds for certain more general α-cosine transforms and for the Radon transform between grassmannians, and more generally for various GLn(R)-modules. These results are then applied to convex geometry and geometric tomography, where sharper versions of the Aleksandrov projection theorem, Funk section theorem, and Klain's and Schneider's injectivity theorems for convex valuations are obtained.
AB - We investigate the support of a distribution f on the real grassmannian Grk(Rn) whose spectrum, namely its nontrivial O(n)-components, is restricted to a subset Λ of all O(n)-types. We prove that unless Λ is co-sparse, f cannot be supported at a point. We utilize this uncertainty principle to prove that if 2≤k≤n−2, then the cosine transform of a distribution on the grassmannian cannot be supported inside any single open Schubert cell Σk. The same holds for certain more general α-cosine transforms and for the Radon transform between grassmannians, and more generally for various GLn(R)-modules. These results are then applied to convex geometry and geometric tomography, where sharper versions of the Aleksandrov projection theorem, Funk section theorem, and Klain's and Schneider's injectivity theorems for convex valuations are obtained.
KW - Cosine transform
KW - Geometric tomography
KW - Quasianalytic
KW - Radon transform
KW - Uncertainty principle
KW - Valuations
UR - http://www.scopus.com/inward/record.url?scp=85175251178&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2023.109348
DO - 10.1016/j.aim.2023.109348
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AN - SCOPUS:85175251178
SN - 0001-8708
VL - 435
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 109348
ER -