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Four short stories about Toeplitz matrix calculations

Thomas Strohmer

TL;DR

The paper addresses solving Toeplitz‑type systems across biinfinite, infinite, and finite settings under off‑diagonal decay, using a blend of Demko–DMS–Smith inverse‑decay results and Wiener–type analysis to characterize when inverses preserve or degrade decay. It derives explicit error bounds for finite‑section approximations in both exponential and polynomial decay regimes, showing convergence rates that depend only on decay and the matrix condition number. For deconvolution problems, it analyzes FFT‑based embedding of Toeplitz into circulant matrices, establishing sharp exponential or polynomial error bounds for the approximate solutions obtained from circulant systems. Finally, it provides quantitative preconditioning results via circulant embedding, proving eigenvalue clustering of the preconditioned Toeplitz systems and explaining observed numerical behavior, with practical implications for FFT‑based solvers in signal processing and related applications.

Abstract

The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.

Four short stories about Toeplitz matrix calculations

TL;DR

The paper addresses solving Toeplitz‑type systems across biinfinite, infinite, and finite settings under off‑diagonal decay, using a blend of Demko–DMS–Smith inverse‑decay results and Wiener–type analysis to characterize when inverses preserve or degrade decay. It derives explicit error bounds for finite‑section approximations in both exponential and polynomial decay regimes, showing convergence rates that depend only on decay and the matrix condition number. For deconvolution problems, it analyzes FFT‑based embedding of Toeplitz into circulant matrices, establishing sharp exponential or polynomial error bounds for the approximate solutions obtained from circulant systems. Finally, it provides quantitative preconditioning results via circulant embedding, proving eigenvalue clustering of the preconditioned Toeplitz systems and explaining observed numerical behavior, with practical implications for FFT‑based solvers in signal processing and related applications.

Abstract

The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.

Paper Structure

This paper contains 5 sections, 11 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. On the decay of inverses of Toeplitz-type matrices
  3. Approximation of infinite-dimensional Toeplitz-type systems
  4. Error estimates for approximate solution of deconvolution problems
  5. Preconditioning by embedding and exponentially decaying Toeplitz matrices

Key Result

Theorem 1.1

Let $A$ be a matrix acting on ${{\boldsymbol\ell}^2}({\@fontswitch{}{\mathcal{}} I})$, where ${\@fontswitch{}{\mathcal{}} I}=\{0,1,\dots,N-1\}, {\mathbb Z}$, or ${\mathbb N}$, and assume $A$ to be hermitian positive definite and $s$-banded (i.e., $A_{kl}=0$ if $|k-l|>s$). Set $\kappa=\|A\| \|A^{-1}\ where

Figures (2)

  • Figure 2.1: Actual approximation error and error estimate \ref{['polest1']} from Theorem \ref{['th1']}(b) for the system $Lx=y$, where $L$ is a biinfinite hermitian Toeplitz matrix with polynomial decay.
  • Figure 3.1: Comparison of error for the solution of a biinfinite Toeplitz system with polynomial decay. We compare the approximation error of the Toeplitz system described in Theorem \ref{['th1']} to that of the circulant system of Corollary \ref{['corcirc']} for increasing matrix dimension. The approximation error of both methods is almost identical, so that the difference between the two graphs is hardly visible.

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • proof
  • Remark 1.5
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.4
  • ...and 13 more