Painless Approximation of Dual Frames, with Applications to Shift-invariant Systems

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