**ABSTRACT**
### Rates of convergence for the approximation of dual
shift-invariant systems in $l^2(Z)$

#### Thomas Strohmer

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
$f = \sum \langle f, \gamma_{m,n} \rangle g_{m,n}$.
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.

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