On the Duality Principle by Casazza, Kutyniok, and Lammers

Abstract

The R-dual sequences of a frame {f(i)}(i is an element of 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 {omega(j)}(j is an element of I) to be an R-dual of a given frame {fi}(i is an element of I). In particular we show that the R-duals {omega(j)}(j is an element of I) can be characterized in terms of frame properties of an associated sequence {n(i)}(i is an element of 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.