Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity

TL;DR

This paper analyzes finite-rank frame operators from cylinder indicator functions on Bernoulli Cantor measures, revealing their structure, self-similarity, and spectral properties, especially through Haar bases and filtration representations.

Contribution

It introduces a new filtration representation for these operators, characterizes the limit operator $K_{ olinebreak} olinebreak_ olinebreak infty$ as self-similar, and derives resolvent formulas, extending understanding of their spectral structure.

Findings

Haar basis diagonalizes operators for p=1/2

Weighted Haar basis yields sparse matrix form for general p

Limit operator $K_{ olinebreak} olinebreak_ olinebreak infty$ is compact, positive, self-adjoint, and self-similar

Abstract

We study the finite-rank frame operators generated by cylinder indicator functions for the Bernoulli Cantor measure $\mu_{p}$. In the symmetric case $p=\frac{1}{2}$, the natural Haar differences diagonalize these operators. For general $0<p<1$, we show that the weighted Haar basis still yields a sparse tree-banded matrix form, although diagonalization is lost. We also prove a filtration representation in terms of conditional expectations and level-wise mass operators. This leads to a norm convergent limit operator $K_{\infty}$, which is compact, positive, and self-adjoint. Finally, we show that $K_{\infty}$ is characterized by a self-similar operator identity induced by the first-level Cantor decomposition, and we derive corresponding block and scalar resolvent renormalization formulas.