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

Thomas Strohmer

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 $\ltZ$ 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 $\LtR$ via truncated Gram matrix.

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