A note on the rate of convergence for a sequence of random polarizations

TL;DR

This paper investigates the rate of convergence of random polarizations of functions and sets to their symmetric counterparts, providing bounds, optimal rates, and introducing Markov chain-based polarization sequences.

Contribution

It extends previous results by establishing new bounds on convergence rates, including for functions in various spaces, and introduces a Markov chain-based polarization sequence.

Findings

Expected $L^1$ distance bounded by $c_n n^{-1}$ with $ar{c}_n$ limit involving $ abla f$

Convergence rate for sets slower than $n^{-r}$ for any $r>2d$

Existence of polarization sequences converging exponentially for sets with finite perimeter

Abstract

It was shown by Burchard and Fortier that the expected $L^1$ distance between $f^*$ and $n$ random polarizations of an essentially bounded function $f$ with support in a ball of radius $L$ is bounded by $2dm(B_{2L})||f||_{\infty}n^{-1}$. This article complements and extends this result. The expected $L^1$ distance is bounded by $c_nn^{-1}$ with $\limsup_{n\rightarrow \infty}c_n \leq 2^{d+1}||\nabla f||_1$ for every $f \in W_{1,1}(B_L) \cap L^{\infty}(B_L)$. Furthermore, the expected $L^1$ distance is $O(n^{-1/q})$ for $f \in L^p(B_L)$ with $p>1$ and $\frac{1}{p} + \frac{1}{q} = 1$. The rate $n^{-1}$ is best possible: $n$ times the measure of the symmetric difference between the random polarizations of a ball and its corresponding Schwarz symmetrization converges in distribution to a random variable with moments that are derived. It is also shown that the expected symmetric difference between the random polarizations of a measurable set in $B_L$ and its corresponding Schwarz symmetrization is slower than $n^{-r}$ for any $r>2d$ and if the rate is $n^{-1}$ then $n$ times the measure of the symmetric difference between the random polarizations of the set and its corresponding Schwarz symmetrization converges in distribution. A new sequence of random polarizations is introduced such that the transition probability depends on the state of the underlying Markov chain. For compact sets with finite perimeter, the rate of convergence is $O(n^{-3/2})$ when $d=1$ and $O(n^{-(1 + \frac{d-1}{d(d+1)})})$ for $d>1$. Finally it is shown that for every compact set $A$ in $\mathbb{R}$ with finite perimeter there exists a sequence of polarizations $A_n$ of $A$ converging exponentially to its Schwarz symmetrization.