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

James Tian

Abstract

We study the finite-rank frame operators generated by cylinder indicator functions for the Bernoulli Cantor measure $μ_{p}$. 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.

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 . In the symmetric case , the natural Haar differences diagonalize these operators. For general , 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 , which is compact, positive, and self-adjoint. Finally, we show that 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.

Paper Structure

This paper contains 9 sections, 23 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cylinders and the Bernoulli measure
  4. Filtration, projections, and frame operators
  5. Symmetric case and Haar structure
  6. Weighted case and filtration
  7. The limit operator $K_{\infty}$ for $p\neq1/2$
  8. Self-similar fixed-point identity
  9. Resolvent of $K_{\infty}$

Key Result

Proposition 3.1

Let $\mathcal{F}_{m}$ be as in eq:2-5. Let Then $\left\{ \phi\right\} \cup\left\{ h_{w}:|w|\le m-1\right\}$ is an orthogonal basis of $\mathcal{F}_{m}$, and the operator $K_{m}$ from eq:2-6--eq:2-7 is diagonal in this basis. More precisely, and if $|w|=\ell$ with $0\le\ell\le m-1$, then Equivalently, if $E_{n}$ denotes the orthogonal projection in eq:2-2--eq:2-3, then as an operator on $L^{2}\

Theorems & Definitions (50)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 40 more