Absolute and Unconditional Convergence of Series of Ergodic Averages and Lebesgue Derivatives

TL;DR

This work investigates when series of averaging operators and their differences converge absolutely or unconditionally in ergodic and harmonic-analytic settings. A central tool is a spectral criterion: unconditional convergence of $\sum\limits_{k} (T_{n_{k+1}}-T_{n_k})f$ for all $f$ in a space like $L^2$ is equivalent to a uniform bound on the cumulative spectral differences $\sum_k |\widehat{T}_{n_{k+1}}(\gamma)-\widehat{T}_{n_k}(\gamma)|$ for the relevant Fourier modes, with parallel statements for probability-measure averages and differentiation operators. The paper provides detailed examples: lacunary sequences (e.g., $n_k=2^k$) yield unconditional convergence in $L^p$ for $1<p<\infty$, while polynomial growth like $n_k=k^p$ with $p>1$ can fail; for circle rotations, generic functions exhibit first-category behavior where absolute convergence is rare, though coboundaries can yield convergence in specific cases. Across $L^2$, $L^2(\mathbb{T})$, and $H_2(\mathbb{T})$, the results connect ergodic theory with harmonic analysis via the differentiation operators $D_{\varepsilon}$ and the spectral symbols $m_n$. Together, these results advance understanding of when averaging-series converge in Banach spaces and illuminate the delicate balance between absolute and unconditional convergence in dynamical and harmonic contexts.

Abstract

We consider when there is absolute or unconditional convergence of series of various types of stochastic processes. These processes include differences of averages in ergodic theory and harmonic analysis, like the classical Cesaro average in ergodic theory and Lebesgue derivatives in harmonic analysis.

Theorems & Definitions (75)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Lemma 2.8
• proof