Sharp Degree Bounds for Signed-Measure Certificates

Overview

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 > exp(cd). Among measures μ = −δ₀ + ν with supp ν contained in {|x| ≥ d}, total variation ‖ν‖ ≤ R, and ∫ exp(c|x|) d|μ| ≤ 1 + CR, let D(R) be the worst least degree of a real polynomial P for which ∫ P² dμ < 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 floor(R). The theorem determines the logarithmic growth exponent, not the optimal logarithmic factor, leading constant, or computational running time.

Keywords signed measures · polynomial certificates · moment positivity · approximation theory · Christoffel function · continuous dual Hahn polynomials