Generalized SHift-Invariant Systems and Dual Frames for Subspaces

Let Tk denote translation by k 2 Zd . Given countable collections of functions fϕjg j2J , fϕ˜jg j2J L2(Rd)and assuming that fTkϕjg j2J;k2Zd and fTk ϕ˜jg j2J;k2Zd are Bessel sequences, we are interested in expansions()Our main result gives an equivalent condition for this to hold in a more general setting than described here,where translation by k 2 Zd is replaced by translation via the action of a matrix. As special cases of our result we find conditions for shift-invariant systems to generate a subspace frame with a corresponding dual having the same structure.In this paper we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames.The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.

Keywords:

Frames for subspaces , generalized shift-invariant systems , approximately dual frames .