Existence of Finite Unit-norm Tight Frames in Banach Spaces
Yam-Sung Cheng and Christopher Heil – GJM, Volume 7, Issue 1 (2022), 17-38.
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- septembre 10, 2022
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A frame provides basis-like but typically nonunique representations of vectors in a Hilbert space. The first part of this paper is a survey of finite unit-norm tight frames, or FUNTFs, in finite-dimensional Hilbert spaces, which were first studied by Benedetto and Fickus. An extension of the concept of FUNTFs from the Hilbert space setting to the Banach space setting was introduced by Chávez–Domínguez, Freeman, and Kornelson, who proved the existence of FUNTFs for complex finite-dimensional Banach spaces, and gave necessary and sufficient conditions for Banach space sequence pairs to be a FUNTF. However, the existence of FUNTFs for real finite-dimensional Banach spaces has been an open question. The second part of this paper makes significant progress towards answering this open question, with explicit constructions of FUNTFs in real n-dimensional space under the \ell_1 -norm. The paper closes with several open questions.
Milestones:
Received: March 12, 2022
Accepted: July 30, 2022
Published online: September 19, 2022
Authors:
Y-S. Cheng and C. HeilSchool of Mathematics,
Georgia Institute of Technology,
Atlanta, Georgia 30332-0160, USA