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Ergodic Estimations for Toeplitz Sequences Generated by a Symbol

Giovanni Barbarino

TL;DR

This paper addresses the rate at which the ergodic formula for Toeplitz matrix sequences generated by a symbol converges at finite sizes. Building on Widom’s framework, it connects the asymptotic singular value distribution to a symbol and uses Fourier-moment estimates and Hankel operator techniques to derive an explicit finite-$n$ bound. The main result (Theorem 1) provides a concrete bound on the deviation $\left|\sum_{j=1}^n G(\sigma_j(T_n(f))^2) - \frac{n}{2\pi}\int_{-\pi}^{\pi} G(|f(\theta)|^2)\,d\theta\right|$, in terms of the symbol norm $|||f|||^2$, the $L^\infty$-norm of $f$, and derivatives of the test function $G$, namely $c_1$ and $c_2$ from explicit formulas. This yields a practical, quantitative rate of convergence for ergodic-type limits in Toeplitz settings and lays groundwork for extending to less regular test functions and broader symbol classes. The work advances spectral analysis of Toeplitz sequences and has potential implications for applications in Bosonic systems and related operator-theoretic problems. The paper also outlines future directions, including handling non-smooth indicators and tightening the bounds further.

Abstract

We analyse the convergence of the ergodic formula for Toeplitz matrix-sequences generated by a symbol and we produce explicit bounds depending on the size of the matrix, the regularity of the symbol and the regularity of the test function.

Ergodic Estimations for Toeplitz Sequences Generated by a Symbol

TL;DR

This paper addresses the rate at which the ergodic formula for Toeplitz matrix sequences generated by a symbol converges at finite sizes. Building on Widom’s framework, it connects the asymptotic singular value distribution to a symbol and uses Fourier-moment estimates and Hankel operator techniques to derive an explicit finite- bound. The main result (Theorem 1) provides a concrete bound on the deviation , in terms of the symbol norm , the -norm of , and derivatives of the test function , namely and from explicit formulas. This yields a practical, quantitative rate of convergence for ergodic-type limits in Toeplitz settings and lays groundwork for extending to less regular test functions and broader symbol classes. The work advances spectral analysis of Toeplitz sequences and has potential implications for applications in Bosonic systems and related operator-theoretic problems. The paper also outlines future directions, including handling non-smooth indicators and tightening the bounds further.

Abstract

We analyse the convergence of the ergodic formula for Toeplitz matrix-sequences generated by a symbol and we produce explicit bounds depending on the size of the matrix, the regularity of the symbol and the regularity of the test function.
Paper Structure (8 sections, 10 theorems, 36 equations)

This paper contains 8 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Known results
  3. Function norms
  4. Estimates for the Fourier transform and series
  5. Toeplitz and Hankel matrices
  6. Widom Estimation
  7. Conclusions and Future Works
  8. Acknowledgement

Key Result

Theorem 1

Let $f\in L^\infty([-\pi,\pi])$ with ${\left\vert\left\vert\left\vert f \right\vert\right\vert\right\vert}^2<\infty$ and let $G\in C^3_c(\mathbb R)$. Then, for every $n\ge 1$, where $c_1= 2\|G'\|_1+\sqrt 2\|G"\|_2$ and $c_2 = 2\|G"\|_1+\sqrt 2\|G"'\|_2$.

Theorems & Definitions (14)

  • Definition 1: asymptotic singular value distribution of a matrix-sequence
  • Theorem 1
  • Theorem 2: Hölder
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • Lemma 3
  • Lemma 4
  • ...and 4 more