Rates of convergence for the approximation of dual shift-invariant systems in $l_2(Z)$
Thomas Strohmer
TL;DR
The paper addresses how to accurately approximate dual shift-invariant systems in $\ell^2(\mathbb{Z})$ using finite-dimensional models. It casts the problem in terms of a frame operator $S$ and its Laurent/operator-symbol structure, and applies the finite-section method to obtain convergence of finite-dimensional duals to the true infinite-dimensional duals. The main contributions are explicit rate results: exponential convergence for compactly supported (FIR) analysis filters, with concrete constants; transfer of decay properties from $g_m$ to the duals under exponential and polynomial decay, and extensions to tight frames with a canonical $S^{-1/2}$-based approximation. It also analyzes a periodic model as a structure-preserving alternative and provides explicit convergence bounds, establishing practical guidelines for numerically computing duals in filter banks and related shift-invariant systems. Overall, the work delivers rigorous, quantitative guarantees for finite-dimensional approximations of dual frames in time-discrete settings and highlights when periodic modeling is advantageous.
Abstract
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 $γ_{m,n}(k) = γ_m(k-an)$ such that each function $f$ can be written as $f = \sum < f, γ_{m,n} > g_{m,n}$. 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.