Sparse Averages of Partial Sums of Fourier Series

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.