Research Article · Preprint Mathematics
Observation norms under double-exponential confinement
- Primary subject
- Mathematics
- Subject subfields
- Spectral Theory · Functional Analysis · Number Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
We determine the small-regularization norm of a weighted observation of a positive Fourier-energy operator under double-exponential spatial confinement. The leading term is the logarithm of the logarithm of the inverse regularization, with an explicit constant given by the quadratic part of the symbol and a constrained Dirichlet eigenvalue on an interval. Finitely many fixed nonzero polynomial-exponential constraint chains leave this constant unchanged; zero-exponent moment chains survive in the limiting problem. In particular a mean-zero constraint halves the leading observation norm. For the original two-row theta-weighted reference, the result gives the exact constant together with finite explicit bounds. Applications include a full factored-residual covariance enclosure and a complete-arithmetic sandwich bound. The proofs retain the specified compact-core closure and all mixed residual terms.
Keywords observation norms · double-exponential confinement · constrained Dirichlet eigenvalues · Fourier energy · theta kernels · residual covariance
Scope of the work
Double-exponential confinement gives a sharp log-log small-regularization weighted observation norm governed by a constrained interval eigenvalue; fixed nonzero exponential chains preserve its constant and zero-exponent moment chains modify it.
Reproduction notes
The 8-page manuscript and complete editable LaTeX sources are supplied in this 7-member package. README.md gives the build and reproduction commands. Reproducibility code: verify_bounds.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.
The manuscript
Read the paper
The complete manuscript, as published in this edition.
If the reader is unavailable on your device, open or download the PDF directly.
Author disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs.
References and related publications
- D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), 906–931.
- M. Demuth, F. Jeske and W. Kirsch, Quantitative estimates for Schrödinger and Dirichlet semigroups (1992).
- J. D. Rossi, The first eigenvalue for nonlocal operators (2022), 741–772.
- S. Schubr, Frequency Localization for Weil Operator Certificates, edition 2 (2026).
- S. Schubr, Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators (2026).
- NIST Digital Library of Mathematical Functions, Chapter 5.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Observation norms under double-exponential confinement (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/observation-norms-double-exponential-confinement/v/2
BibTeX
@techreport{luminary_observation_norms_double_exponential_confinement_v2,
title = {Observation norms under double-exponential confinement},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/observation-norms-double-exponential-confinement/v/2}
}Loading discussion…