Almost-Orthogonality of Restricted Haar-Functions

Abstract

We consider the Haar functions $h_I$ on dyadic intervals. We show that if $p>\frac{2}{3}$ and $E\subset [0,1]$ then the set of all functions $\Vert h_I{1}_E{\Vert }_2^{-1}{h}_I{1}_E$ with $|I\cap E|\ge p|I|$ is a Riesz sequence. For $p\le \frac{2}{3}$ we provide a counterexample.