On the Duality Principle by Casazza, Kutyniok, and Lammers

摘要

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.