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 n0<n1<n2<... and looking at their average to determine the necessary conditions on the sequence {nk} 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, nk=pk, for p a positive integer, and present strong experimental evidence for convergence of the quadratic nk=k2 and cubic nk=k3 cases, but divergence for the exponential case, nk=2k. 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.

For full paper and abstract, see https://arxiv.org/abs/1807.07636.

The files here reflect the MATLAB based generators for the graphs seen in the above abstract and paper.

Created by Ethan Goolish with advisor Robert S. Strichartz, Cornell University Department of Mathematics.