Almost-Orthogonality of Restricted Haar-Functions

TL;DR

This paper investigates the near-orthogonality properties of restricted Haar functions on dyadic intervals, establishing conditions under which they form a Riesz sequence and providing counterexamples for other cases.

Contribution

It introduces a threshold condition for the almost-orthogonality of restricted Haar functions and demonstrates the boundary case with a counterexample.

Findings

For p > 2/3, the set of normalized restricted Haar functions forms a Riesz sequence.

Counterexamples exist for p ≤ 2/3, showing the limit of the Riesz sequence property.

The result clarifies the structure of Haar functions restricted to subsets of [0,1].

Abstract

We consider the Haar functions $h_I$ on dyadic intervals. We show that if $p>\frac23$ and $E\subset[0,1]$ then the set of all functions $\|h_I1_E\|_2^{-1}h_I1_E$ with $|I\cap E|\geq p|I|$ is a Riesz sequence. For $p\leq\frac23$ we provide a counterexample.