  A shift-invariant system is a collection of functions $\{g_{m,n}\}$ of the
form $g_{m,n}(k) = g_m(k-an)$. Such systems play an important role in
time-frequency analysis and digital signal processing. A principal problem is
to find a dual system $\gamma_{m,n}(k) = \gamma_m(k-an)$ such that each
function $f$ can be written as $f = \sum < f, \gamma_{m,n} > g_{m,n}$. The
mathematical theory usually addresses this problem in infinite dimensions
(typically in $L_2(R)$ or $l_2(Z)$), whereas numerical methods have to operate
with a finite-dimensional model. Exploiting the link between the frame operator
and Laurent operators with matrix-valued symbol, we apply the finite section
method to show that the dual functions obtained by solving a finite-dimensional
problem converge to the dual functions of the original infinite-dimensional
problem in $l_2(Z)$. For compactly supported $g_{m,n}$ (FIR filter banks) we
prove an exponential rate of convergence and derive explicit expressions for
the involved constants. Further we investigate under which conditions one can
replace the discrete model of the finite section method by the periodic
discrete model, which is used in many numerical procedures. Again we provide
explicit estimates for the speed of convergence. Some remarks on tight frames
complete the paper.

Numerical Analysis