Research Article · Preprint Mathematics
Polynomial inertia of signed measures with negative intervals
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis · Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords signed measures · polynomial inertia · negative intervals · Meixner-Pollaczek polynomials · exponential moments
Scope of the work
Logarithmically growing negative polynomial inertia for signed measures negative on intervals under an absolute exponential moment, with sharp infimum coefficient and stronger-tail extensions.
Reproduction notes
11-page manuscript and 8 source-package members. Use python -B build.py with an installed Tectonic. Standard-library Python runs verify_mp_grams.py and verify_inertia_independent.py, also under -O, for separate exact congruence and characteristic-polynomial inertia controls.
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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.
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Cite this edition
Scott Schubr. (2026). Polynomial inertia of signed measures with negative intervals (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/polynomial-inertia-signed-measures-negative-intervals/v/2
BibTeX
@techreport{luminary_polynomial_inertia_signed_measures_negative_intervals_v2,
title = {Polynomial inertia of signed measures with negative intervals},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/polynomial-inertia-signed-measures-negative-intervals/v/2}
}Loading discussion…