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Observation norms under double-exponential confinement

Scott Schubr

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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.

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Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-26 · public

Research details

Subjects
Mathematics · Number Theory · Functional Analysis · Spectral Theory
Keywords
observation norms, double-exponential confinement, constrained Dirichlet eigenvalues, Fourier energy, theta kernels, residual covariance
Manuscript date
2026-10-26
Reader reuse
No additional permission granted

Scope

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.

Disclosures

AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs.

Reproduction

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.

References and related resources

  1. D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), 906–931.
  2. M. Demuth, F. Jeske and W. Kirsch, Quantitative estimates for Schrödinger and Dirichlet semigroups (1992).
  3. J. D. Rossi, The first eigenvalue for nonlocal operators (2022), 741–772.
  4. S. Schubr, Frequency Localization for Weil Operator Certificates, edition 2 (2026).
  5. S. Schubr, Joint Cutoffs and Negative-Spectrum Decay for Weighted Weil Operators (2026).
  6. NIST Digital Library of Mathematical Functions, Chapter 5.