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Rates of convergence for the approximation of dual shift-invariant systems in ℓ2(ℤ)

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 95616-8633, Davis, CA

    Thomas Strohmer

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  1. Thomas Strohmer
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Communicated by Richard Tolimieri

Part of this work was done while the author was a visitor at the Department of Statistics at the Stanford University.

The author has been partially supported by Erwin-Schrödinger scholarship J01388-MAT of the Austrian Science foundation FWF.

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Strohmer, T. Rates of convergence for the approximation of dual shift-invariant systems in ℓ2(ℤ). The Journal of Fourier Analysis and Applications 5, 599–615 (1999). https://doi.org/10.1007/BF01257194

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