Rates of convergence for the approximation of dual shift-invariant systems in ?2(?)

Abstract

A shift-invariant system is a collection of functions {gm,n} of the form gm,n(k)=gm(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 γm,n(k)=γm(k−an) such that each functionf can be written asf= ∑〈f, γm,n〉gm,n. The mathematical theory usually addresses this problem in infinite dimensions (typically in L2 (ℝ) or l2(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 l2(ℤ). For compactly supported gm, 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.