On the Duality Principle by Casazza, Kutyniok, and Lammers
Publication: Research - peer-review › Journal article – Annual report year: 2011
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On the Duality Principle by Casazza, Kutyniok, and Lammers. / Christensen, Ole; Kim, Hong Oh; Kim, Rae Young.
In: Journal of Fourier Analysis and Applications, Vol. 17, No. 4, 2011, p. 640-655.Publication: Research - peer-review › Journal article – Annual report year: 2011
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TY - JOUR
T1 - On the Duality Principle by Casazza, Kutyniok, and Lammers
A1 - Christensen,Ole
A1 - Kim,Hong Oh
A1 - Kim,Rae Young
AU - Christensen,Ole
AU - Kim,Hong Oh
AU - Kim,Rae Young
PY - 2011
Y1 - 2011
N2 - The R-dual sequences of a frame {f i } i∈I , introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ω j } j∈I to be an R-dual of a given frame {f i } i∈I . In particular we show that the R-duals {ω j } j∈I can be characterized in terms of frame properties of an associated sequence {n i } i∈I . We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.
AB - The R-dual sequences of a frame {f i } i∈I , introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ω j } j∈I to be an R-dual of a given frame {f i } i∈I . In particular we show that the R-duals {ω j } j∈I can be characterized in terms of frame properties of an associated sequence {n i } i∈I . We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.
KW - Wexler-Raz theorem
KW - Gabor system
KW - Frame
KW - Riesz basis
KW - Duality principle
U2 - 10.1007/s00041-010-9151-4
DO - 10.1007/s00041-010-9151-4
JO - Journal of Fourier Analysis and Applications
JF - Journal of Fourier Analysis and Applications
SN - 10695869
IS - 4
VL - 17
SP - 640
EP - 655
ER -