Polynomial inertia of signed measures with negative intervals
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Abstract
Let μ be a finite real signed measure with a finite absolute exponential moment at rate c>0. Suppose its restriction to each of finitely many disjoint bounded intervals is at most a strictly negative constant times Lebesgue measure. On the real polynomials of degree at most N, we prove that the negative index of the original square-integral form is at least (cL/π^2+o(1))log N, where L is the total interval length. The coefficient is optimal over this class, as an infimum, by a classical Meixner–Pollaczek model. Arbitrary signed and singular exterior mass is allowed. The construction preserves the whole negative span, permits rational polynomial bases, and gives the corresponding bound in each parity for even measures. An absolute exp(c|x|^p) moment, p>1, instead gives an explicit lower bound of order N^(1-1/p); its power is optimal for p=2. A credited Fourier-gap construction explains why p<1 can allow a negative interval invisible to every polynomial square. Finally, a conditional application to the original positive-width Riemann-theta form converts a supplied negative interval, or a hypothetical off-axis zero with local residual bounds, into logarithmically growing negative polynomial inertia.
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory · Classical and Real Analysis
- Keywords
- signed measures, negative index, polynomial approximation, exponential moments, Meixner-Pollaczek polynomials
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Logarithmically growing negative polynomial inertia for signed measures negative on intervals under an absolute exponential moment, with sharp infimum coefficient and stronger-tail extensions.
Disclosures
AI tools assisted with mathematical development, proof checking and manuscript preparation. The author is responsible for the statements and proofs. This preprint does not claim external peer review or formal proof verification.
Reproduction
12-page manuscript and eight-member source package. The editable LaTeX source includes the bibliography. Build with build.py and Tectonic; README.md gives ordinary and optimized commands for both standard-library finite inertia verifiers. Their 65 full/130 parity cases and 50 independent controls corroborate the stated finite model identities. The general lower laws and conditional original-theta application have analytic proofs. CITATION.cff and MANIFEST.sha256 supply citation and exact member identities.