Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913006995111936 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04257 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity Tian, James Functional Analysis Primary: 42C40, Secondary: 28A80, 42C15, 46L05, 47A10 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. |
| title | Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity |
| topic | Functional Analysis Primary: 42C40, Secondary: 28A80, 42C15, 46L05, 47A10 |
| url | https://arxiv.org/abs/2604.04257 |