Painless approximation of dual frames, with applications to shift-invariant systems

We analyse the relation between infinite-dimensional frame theory and finite-dimensional models for frames as they are used for numerical algorithms. Special emphasis in this paper is on perfect reconstruction oversampled filter banks, also known as shift-invariant frames. For certain finite-dimensional models it is shown that the corresponding finite dual frame provides indeed an approximation of the dual frame of the original infinite-dimensional dual frame. For filter banks on l(2)(Z) we derive error estimates for the approximation of the synthesis filter bank when the analysis filter bank satisfies certain decay conditions. We show how one has to design the finite-dimensional model to preserve important structural properties of filter banks, such as polyphase representation. Finally an efficient regularization method is presented to solve the ill-posed problem arising when approximating the dual frame on L-2(R) via truncated Gram matrix.