Sparse Averages of Partial Sums of Fourier Series

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 $n_0<n_1<n_2<\mathrm{.}\mathrm{.}\mathrm{.}$ 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.