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Sparse Averages of Partial Sums of Fourier Series
Ethan Goolish, Robert S. Strichartz
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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 σ̃_N(f)(t) = 1/(N+1)∑ₖ₌₀ᴺs_(nₖ)(f)(t) 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.
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