Observation norms under double-exponential confinement
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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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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
The sharp law concerns the stated two-sided double-exponential wall, quadratic-at-zero symbol, fixed finite constraint chains and specified compact-core closure. The scalar/Taylor comparison in Appendix A concerns its displayed sufficient design.
Disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs.
Reproduction
The publication package contains the complete LaTeX source and a portable python-flint 0.9.0 verifier. Tectonic 0.17.0 builds the manuscript. The verifier reconstructs the finite observation caps and both scalar symbol certificates using directed Arb/Acb arithmetic; README.md gives the commands.
References and related resources
- 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.