Postorder rearrangement operators

TL;DR

This paper studies how the Haar system rearranged by postorder differs from the standard order, showing it has maximal distance in operator norm within the BMO space.

Contribution

It introduces a new analysis of the postorder rearrangement of the Haar system and quantifies its maximal deviation from the lexicographic order using operator norms.

Findings

Postorder rearrangement has maximal distance to lexicographic order in BMO norms.

Operator norms on $ ext{BMO}_N$ used to measure rearrangement differences.

Rearrangement properties depend on dyadic interval structure.

Abstract

We investigate the rearrangement of the Haar system induced by the postorder on the set of dyadic intervals in $[0,1]$ with length greater than or equal to $2^{-N}$. By means of operator norms on $\text{BMO}_N$ we prove that the postorder has maximal distance to the usual lexicographic order.