  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.
