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A new identity for Parseval frames
Authors:
Radu Balan, Peter G. Casazza, Dan Edidin and Gitta Kutyniok
Journal:
Proc. Amer. Math. Soc. 135 (2007), 1007-1015
MSC (2000):
Primary 42C15; Secondary 94A12
Posted:
November 14, 2006
MathSciNet review:
2262901
Full-text PDF Free Access
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Additional Information
Abstract: In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.
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- 2.
- R. Balan, P.G. Casazza, D. Edidin, and G. Kutyniok, Decompositions of frames and a new frame identity, Wavelets XI (San Diego, CA, 2005), 379-388, SPIE Proc. 5914, SPIE, Bellingham, WA, 2005.
- 3.
- J.J. Benedetto and M. Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003), 357-385. MR 1968126 (2004c:42059)
- 4.
- P.G. Casazza, The art of frame theory, Taiwanese J. of Math. 4 (2000), 129-201. MR 1757401 (2001f:42046)
- 5.
- P.G. Casazza, Custom building finite frames, in: Wavelets, frames and operator theory, 61-86, Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 2004. MR 2066822 (2005f:42078)
- 6.
- P.G. Casazza and J. Kovacevic, Equal-norm tight frames with erasures, Adv. Comput. Math. 18 (2003), 387-430. MR 1968127 (2004e:42046)
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- P.G. Casazza and G. Kutyniok, A generalization of Gram-Schmidt orthogonalization generating all Parseval frames, Adv. Comput. Math., to appear.
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- R.H. Chan, S.D. Riemenschneider, L. Shen, Z. Shen, Tight frame: an efficient way for high-resolution image reconstruction, Appl. Comput. Harmon. Anal. 17 (2004), 91-115. MR 2067917 (2005h:94006)
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- O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston (2003). MR 1946982 (2003k:42001)
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- Y.C. Eldar and G.D. Forney, Jr., Optimal tight frames and quantum measurement, IEEE Trans. Inform. Theory 48 (2002), 599-610. MR 1889971 (2003c:94006)
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- V.K. Goyal, J. Kovacevic, and J.A. Kelner, Quantized frame expansions with erasures, Appl. Comput. Harmon. Anal. 10 (2001), 203-233. MR 1829801 (2002h:94012)
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- R. Vale and S. Waldron, Tight frames and their symmetries, Constr. Approx. 21 (2005), 83-112. MR 2105392 (2005h:42063)
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Additional Information
Radu Balan
Affiliation:
Siemens Corporate Research, 755 College Road East, Princeton, New Jersey 08540
Email:
radu.balan@siemens.com
Peter G. Casazza
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email:
pete@math.missouri.edu
Dan Edidin
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email:
edidin@math.missouri.edu
Gitta Kutyniok
Affiliation:
Institute of Mathematics, Justus-Liebig-University Giessen, 35392 Giessen, Germany
Email:
gitta.kutyniok@math.uni-giessen.de
DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08930-1
PII:
S 0002-9939(06)08930-1
Keywords:
Bessel sequence,
frame,
Hilbert space,
Parseval frame,
Parseval Frame Identity
Received by editor(s):
June 13, 2005
Posted:
November 14, 2006
Additional Notes:
The second author was supported by NSF DMS 0405376.
The third author was supported by NSA MDA 904-03-1-0040.
The fourth author was supported by DFG research fellowship KU 1446/5.
Communicated by:
Michael T. Lacey
Article copyright:
© Copyright 2006 American Mathematical Society
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