Sharp degree exponents for polynomial detection of isolated negative atoms
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Abstract
We study how much polynomial degree can be required to detect an isolated negative atom of a signed measure by a negative square integral. Fix a gap d>0, an exponential-moment rate c>0, and C>e^(cd). Among measures μ=-δ_0+ν with suppν⊂{|x|≥ d}, ‖ν‖_(TV)≤ R, and ∫ e^(c|x|)d|μ|≤1+CR, let D(R) be the worst least degree of a real polynomial P for which ∫ P^2dμ<0. We prove log D(R)/log R→π/(2cd), with the explicit-order upper bound D(R)=O(R^(π/(2cd))log(2+R)). A direct critical-weight approximation of a classical continuous Chebyshev window constructs one rational polynomial with negative square integral on the original measure, without evaluating its moments. Continuous dual Hahn laws give matching lower exponents while keeping all input budgets fixed. For finitely many isolated negative atoms we also recover a whole negative polynomial subspace with an explicit polynomial cost in inverse separation; the index is exact for a positive remainder and is a lower bound for a signed remainder. An exact infinite-support example has first negative degree 2⌊ R⌋. The theorem determines the logarithmic growth exponent.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis
- Keywords
- signed measures, polynomial certificates, moment positivity, approximation theory, Christoffel function, continuous dual Hahn polynomials
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
The sharp logarithmic degree exponent for detecting an isolated negative signed-measure atom under fixed exponential-moment and gap budgets, with multi-atom negative-subspace bounds.
Disclosures
AI tools assisted mathematical development, drafting, literature comparison and independent internal checks. The author is responsible for the statements and presentation. These checks are not external human refereeing or formal proof-assistant verification.
Reproduction
11-page manuscript and 9 source-package members. Compile main.tex. Python 3.10+ runs verify.py --certificate certificate.json, verify_independent.py --certificate certificate.json and verify_cardinal.py, also under -O; these check the exact moment, recurrence, null-boundary and mixed-Gram controls.
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