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 -