Abstract
This chapter presents an overview of close parallels that exist between the theory of positive-operator-valued measures (POVMs) associated with a separable Hilbert space and the theory of frames on that space, including its most important generalizations. The concept of a framed POVM is introduced, and classical frames, fusion frames, generalized frames, and other variants of frames are all shown to arise as framed POVMs. This observation allows drawing on a rich existing theory of POVMs to provide new perspectives in the study of frames.
| Original language | English (US) |
|---|---|
| Title of host publication | Applied and Numerical Harmonic Analysis |
| Publisher | Springer International Publishing |
| Pages | 49-64 |
| Number of pages | 16 |
| Edition | 9780817683788 |
| DOIs | |
| State | Published - Jan 1 2013 |
Publication series
| Name | Applied and Numerical Harmonic Analysis |
|---|---|
| Number | 9780817683788 |
| ISSN (Print) | 2296-5009 |
| ISSN (Electronic) | 2296-5017 |
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Keywords
- Frame
- Framed POVM
- Fusion frame
- Generalized frame
- Naimark’s Theorem
- Positive operator-valued measure(POVM)
- Radon-Nikodym Theorem
- Spectral measure
- Stinespring’s Theorem
ASJC Scopus subject areas
- Applied Mathematics
Cite this
Positive-operator-valued measures : A general setting for frames. / Moran, Bill; Howard, Stephen; Cochran, Douglas.
Applied and Numerical Harmonic Analysis. 9780817683788. ed. Springer International Publishing, 2013. p. 49-64 (Applied and Numerical Harmonic Analysis; No. 9780817683788).Research output: Chapter in Book/Report/Conference proceeding › Chapter
}
TY - CHAP
T1 - Positive-operator-valued measures
T2 - A general setting for frames
AU - Moran, Bill
AU - Howard, Stephen
AU - Cochran, Douglas
PY - 2013/1/1
Y1 - 2013/1/1
N2 - This chapter presents an overview of close parallels that exist between the theory of positive-operator-valued measures (POVMs) associated with a separable Hilbert space and the theory of frames on that space, including its most important generalizations. The concept of a framed POVM is introduced, and classical frames, fusion frames, generalized frames, and other variants of frames are all shown to arise as framed POVMs. This observation allows drawing on a rich existing theory of POVMs to provide new perspectives in the study of frames.
AB - This chapter presents an overview of close parallels that exist between the theory of positive-operator-valued measures (POVMs) associated with a separable Hilbert space and the theory of frames on that space, including its most important generalizations. The concept of a framed POVM is introduced, and classical frames, fusion frames, generalized frames, and other variants of frames are all shown to arise as framed POVMs. This observation allows drawing on a rich existing theory of POVMs to provide new perspectives in the study of frames.
KW - Frame
KW - Framed POVM
KW - Fusion frame
KW - Generalized frame
KW - Naimark’s Theorem
KW - Positive operator-valued measure(POVM)
KW - Radon-Nikodym Theorem
KW - Spectral measure
KW - Stinespring’s Theorem
UR - http://www.scopus.com/inward/record.url?scp=85047382621&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85047382621&partnerID=8YFLogxK
U2 - 10.1007/978-0-8176-8379-5_4
DO - 10.1007/978-0-8176-8379-5_4
M3 - Chapter
T3 - Applied and Numerical Harmonic Analysis
SP - 49
EP - 64
BT - Applied and Numerical Harmonic Analysis
PB - Springer International Publishing
ER -