Polynomial Recovery and Masking under Weak Exponential Tails
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Abstract
We determine the worst polynomial degree needed to detect an isolated negative atom in a signed measure under coarse local-mass and weak exponential-moment bounds. For a tail scale H(x)=x/W(x), with W continuous, nondecreasing and subject to a sublinear power-ratio bound, let Ψ(t)=1+∫_1^td x/[xW(x)]. If this clock diverges, the minimax detecting degree satisfies Ψ(D_W(R))=Θ(R^(1/q)) for every fixed local exponent q>0. If it converges, one fixed atomless reference law, scaled with R, masks the negative atom from every polynomial square. No differentiability or regular variation is required. A finite positive moment realization and classical Bang propagation give the uniform signed-measure upper bound. A compact polynomial multiplier and singular Jacobi evaluation give matching fixed-law lower bounds and masking. Logarithmic and iterated-logarithmic tail regimes follow as applications, including double- and triple-exponential critical degrees. We distinguish the original signed-class recovery theorem from the classical determinacy thresholds and approximation ingredients used in its proof.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis · Functional Analysis
- Keywords
- signed measures, polynomial recovery, weak exponential tails, quasianalyticity, moment problems, polynomial masking
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
For continuous nondecreasing tail profiles satisfying the sublinear power-ratio bound, the integrated clock characterizes minimax polynomial detection degree in the divergent regime and supplies fixed-law all-degree masking in the convergent regime.
Disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author declares no competing interests.
Reproduction
The 13-page manuscript and complete editable LaTeX sources are supplied in this 13-member package. README.md gives the build and reproduction commands. Reproducibility code: verify_finite_controls.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.