The Square Variation of Rearranged Fourier Series

TL;DR

This paper demonstrates that rearranging the Fourier series elements can significantly reduce the square variation operator's norm, improving upon the canonical ordering's bounds.

Contribution

It introduces a novel rearrangement method for Fourier series that achieves a lower bound on the square variation operator's norm.

Findings

Rearranged Fourier series can have a square variation norm of at most O(log^{9/22+ε}(N)).

Improves previous bound of O(log^{1/2}(N)) for the canonical ordering.

Provides a new approach to controlling variation in Fourier analysis.

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.