The Square Variation of Rearranged Fourier Series

Abstract

We prove that there exists a rearrangement of the first $N$ elements of the trigonometric system such that the $L^2$-norm of the square variation operator is at most $O_{\epsilon }({\log }^{9/22+\epsilon }(N))$. This is an improvement over $O({\log }^{1/2}(N))$ from the canonical ordering.

27 pages