Sparse Averages of Partial Sums of Fourier Series

TL;DR

This paper investigates the convergence of sparse averages of Fourier partial sums for continuous functions, identifying conditions on the sequence of indices that ensure uniform convergence, with theoretical and experimental results for various sequences.

Contribution

It provides new convergence results for specific sequences like linear, quadratic, and cubic, and offers experimental evidence for random sequences, expanding understanding of Fourier series behavior.

Findings

Convergence for linear sequences $n_k=pk$.

Experimental evidence of convergence for quadratic and cubic sequences.

Divergence observed for exponential sequences $n_k=2^k$.

Abstract

We study convergence properties of sparse averages of partial sums of Fourier series of continuous functions. By sparse averages, we are considering an increasing sequences of integers $n_0 < n_1 < n_2 < ...$ and looking at \begin{equation*} \tilde{\sigma}_N(f)(t) = \frac{1}{N+1}\sum_{k=0}^{N}s_{n_k}(f)(t) \end{equation*} to determine the necessary conditions on the sequence $\{n_k\}$ for uniform convergence. Among our results, we find that convergence is dependent on the sequence: we give a proof of convergence for the linear case, $n_k = pk$, for $p$ a positive integer, and present strong experimental evidence for convergence of the quadratic $n_k = k^2$ and cubic $n_k = k^3$ cases, but divergence for the exponential case, $n_k = 2^k$. We also present experimental evidence that if we replace the deterministic rules above by random processes with the same asymptotic behavior then almost surely the answer is the same.