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Quantitative uniform exponential acceleration of averages along decaying waves

Zhicheng Tong, Yong Li

TL;DR

This work shows that weighted Birkhoff averages with an exponential weighting function achieve uniform exponential convergence along decaying waves, extending beyond quasi-periodic contexts to general decaying modes and nonlinear systems with negative Lyapunov exponents. The authors develop a robust Fourier-analytic approach, including Poisson summation and sharp bounds on derivatives of weighting functions, to derive explicit rates: the weighted decaying-wave averages decay like exp(-xi sqrt(N)) with xi > 0, and unweighted averages exhibit only polynomial rates in most cases. They further generalize to broader weight families w_p,q and widen the applicability to continuous-time settings and to analytic quasi-periodic dynamics, yielding precise quantitative rates under various regularity and nonresonance conditions. Numerical simulations corroborate the sharpness and uniformity of the rates, and the results provide practical acceleration for high-precision computations in dynamical systems, as well as insights into how weight choices influence convergence guarantees across different dynamical regimes.

Abstract

In this study, utilizing a specific exponential weighting function, we investigate the uniform exponential convergence of weighted Birkhoff averages along decaying waves and delve into several related variants. A key distinction from traditional scenarios is evident here: despite reduced regularity in observables, our method still maintains exponential convergence. In particular, we develop new techniques that yield very precise rates of exponential convergence, as evidenced by numerical simulations. Furthermore, this innovative approach extends to quantitative analyses involving different weighting functions employed by others, surpassing the limitations inherent in prior research. It also enhances the exponential convergence rates of weighted Birkhoff averages along quasi-periodic orbits via analytic observables. To the best of our knowledge, this is the first result on the uniform exponential acceleration beyond averages along quasi-periodic or almost periodic orbits, particularly from a quantitative perspective.

Quantitative uniform exponential acceleration of averages along decaying waves

TL;DR

This work shows that weighted Birkhoff averages with an exponential weighting function achieve uniform exponential convergence along decaying waves, extending beyond quasi-periodic contexts to general decaying modes and nonlinear systems with negative Lyapunov exponents. The authors develop a robust Fourier-analytic approach, including Poisson summation and sharp bounds on derivatives of weighting functions, to derive explicit rates: the weighted decaying-wave averages decay like exp(-xi sqrt(N)) with xi > 0, and unweighted averages exhibit only polynomial rates in most cases. They further generalize to broader weight families w_p,q and widen the applicability to continuous-time settings and to analytic quasi-periodic dynamics, yielding precise quantitative rates under various regularity and nonresonance conditions. Numerical simulations corroborate the sharpness and uniformity of the rates, and the results provide practical acceleration for high-precision computations in dynamical systems, as well as insights into how weight choices influence convergence guarantees across different dynamical regimes.

Abstract

In this study, utilizing a specific exponential weighting function, we investigate the uniform exponential convergence of weighted Birkhoff averages along decaying waves and delve into several related variants. A key distinction from traditional scenarios is evident here: despite reduced regularity in observables, our method still maintains exponential convergence. In particular, we develop new techniques that yield very precise rates of exponential convergence, as evidenced by numerical simulations. Furthermore, this innovative approach extends to quantitative analyses involving different weighting functions employed by others, surpassing the limitations inherent in prior research. It also enhances the exponential convergence rates of weighted Birkhoff averages along quasi-periodic orbits via analytic observables. To the best of our knowledge, this is the first result on the uniform exponential acceleration beyond averages along quasi-periodic or almost periodic orbits, particularly from a quantitative perspective.
Paper Structure (14 sections, 87 equations, 2 figures)

This paper contains 14 sections, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['SWT1']}
  3. Proof of (I): Exponential convergence of the weighted type
  4. Proof of (II): $\mathcal{O}\left(N^{-1}\right)$ convergence of the unweighted type
  5. Proof of Corollary \ref{['DWCORO11']}
  6. Proof of Theorem \ref{['DWCORO2']}
  7. Further discussions on general exponential weighting functions
  8. Exponential weighting function with two regularity parameters $p,q$
  9. Exponential weighting function with a width parameter $\gamma$
  10. Further applicability: the continuous case and the general decaying case
  11. The continuous case
  12. The general decaying case
  13. Exponential acceleration in analytic quasi-periodic dynamical systems revisited
  14. Numerical simulation and analysis of convergence rates

Figures (2)

  • Figure 1: Comparison of the convergence rates
  • Figure 2: Comparison of the theoretical error with the actual error

Theorems & Definitions (8)

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