Polynomial inertia of signed measures with negative intervals

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. A finite absolute Gaussian moment gives the second sharp coefficient L√(2c)/π on the √N scale. More generally, an absolute exp(c|x|^p) moment, p>1, gives a lower bound of order N^(1-1/p), with optimal power for every p>1. 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.