Rates of convergence for the approximation of dual shift-invariant systemsin l(2)(Z)

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-fre quency analysis and digital signal processing. A principal problem is to fi nd a dual system gamma(m,n) (k) = gamma(m)(k - an) such that each function f can be written as f = Sigma (f, gamma(m,n))g(m,n). The mathematical theor y usually addresses this problem in infinite dimensions (typically in L-2(R ) or e(2)(Z)), whereas numerical methods have to operate with a finite-dime nsional model. Exploiting the link between the frame operator and Laurent o perators with matrix-valued symbol, we apply the finite section method to s how that the dual functions obtained by solving a finite-dimensional proble m converge to the dual functions of the original infinite-dimensional probl em in e(2)(Z). For compactly supported g(m.n) (FIR filter banks) we prove a n exponential rate of convergence and derive explicit expressions for the i nvolved constants. Further we investigate under which conditions one can re place the discrete model of the finite section method by the periodic discr ete model, which is used in many numerical procedures. Again we provide exp licit estimates for the speed of convergence. Some remarks on tight frames complete the paper.