Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Frames generated by compact group actions


Author: Joseph W. Iverson
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 42C15, 43A77, 47A15; Secondary 22D10, 43A32
DOI: https://doi.org/10.1090/tran/6954
Published electronically: August 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a compact group, and let $ \rho $ be a representation of $ K$ on a Hilbert space $ \mathcal {H}_\rho $. We classify invariant subspaces of $ \mathcal {H}_\rho $ in terms of range functions, and investigate frames of the form $ \{\rho (\xi ) f_i\}_{\xi \in K, i \in I}$. This is done first in the setting of translation invariance, where $ K$ is contained in a larger group $ G$ and $ \rho $ is left translation on $ \mathcal {H}_\rho = L^2(G)$. For this case, our analysis relies on a new, operator-valued version of the Zak transform. For more general representations, we develop a calculational system known as a bracket to analyze representation structures and frames with a single generator. Several applications are explored. Then we turn our attention to frames with multiple generators, giving a duality theorem that encapsulates much of the existing research on frames generated by finite groups, as well as classical duality of frames and Riesz sequences.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42C15, 43A77, 47A15, 22D10, 43A32

Retrieve articles in all journals with MSC (2010): 42C15, 43A77, 47A15, 22D10, 43A32


Additional Information

Joseph W. Iverson
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: jiverson@math.umd.edu

DOI: https://doi.org/10.1090/tran/6954
Keywords: Compact group, frame, invariant subspace, range function, translation-invariant space, unitary representation, Zak transform
Received by editor(s): September 22, 2015
Received by editor(s) in revised form: April 11, 2016
Published electronically: August 15, 2017
Additional Notes: This research was supported in part by NSF grant DMS-1265711, and by Dustin G. Mixon’s AFOSR Young Investigator Research Program award. The views expressed in this article are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.
Article copyright: © Copyright 2017 American Mathematical Society