Polynomial Recovery and Masking under Weak Exponential Tails
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Abstract
We determine the worst polynomial degree needed to detect a negative atom under local variation and weak exponential-moment bounds. For a tail scale H(x)=x/W(x) with a controlled growth ratio, put Ψ(t)=1+∫₁ᵗ dx/[xW(x)]. When Ψ diverges, the worst detecting degree satisfies Ψ(max{1,D_W(R)})=Θ(R^(1/q)) for every fixed local exponent q>0. The same estimates yield the full-line origin Christoffel order λ_n≍Ψ(n)^(-q) for the canonical density proportional to |x|^(q−1)exp(−bH(|x|)). Finite-clock masking and logarithmic critical scales follow within this framework. We also construct a smooth nondecreasing profile with H(x)/log x→∞ and divergent Ψ, for which that canonical density nevertheless masks the atom at every degree. Thus divergent clock and existence of all moments do not replace growth control. The proofs combine classical quasianalytic propagation and localization with explicit moment and interpolation estimates, retaining signed and singular backgrounds and precise fixed-parameter quantifiers.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis · Functional Analysis
- Keywords
- moment problems, Christoffel functions, weighted polynomial approximation, quasianalyticity, signed measures, weak exponential tails
- Manuscript date
- 2026-10-25
- Reader reuse
- none
Scope
The matched integrated-clock detection law holds for continuous nondecreasing W≥1 with W(0)=1 under the sublinear power-ratio bound, for every fixed local exponent q>0. Its canonical-density origin Christoffel estimate and fixed-law masking describe the corresponding extremal behavior. An explicit smooth monotone profile demonstrates all-degree canonical masking despite a divergent clock and existence of all moments when growth control is removed.
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 18-page manuscript and complete editable LaTeX sources are supplied in this 16-member package. README.md gives build and reproduction commands. Two portable standard-library Python programs verify finite Jacobi/kernel controls and finite sparse-interpolation identities. SHA256SUMS.txt identifies every other package member. The analytic proofs establish the all-degree and asymptotic statements.