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

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.