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Growing-Index Recovery and Tail Scales for Signed Measures

Scott Schubr

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Abstract

We determine the minimax polynomial degree needed to recover a growing negative index of a signed measure. For a probability supported in a shrinking interval, let h_m be its degree-m monic squared norm and L = log(R/h_m). Fix a gap d > 0 from that entire interval and a tail rate c > 0. Under the Gaussian background budget with weight exp(cx²), the degree for recovering m+1 negative directions is asymptotic to L²/(8cd²). Under the separately specified full absolute-moment budget with weight exp(c|x|), its logarithm is asymptotic to πL/(2cd). Support concentration alone pays the growing-index costs, including for continuous and singular negative probabilities. Without a gap, a single negative atom and a background local mass bound of order Rr^q give matching recovery orders for uniformly superlinear tail scales and for x[log(e+x)]^β, with fixed β ≥ 0 and q > 0. Adaptive rational damping controls the full growing polynomial span; compact multipliers and a paid variable-rate expansion handle the gapless classes. Classical kernel and orthogonal-polynomial inputs are credited throughout.

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Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-26 · public

Research details

Subjects
Mathematics · Classical and Real Analysis
Keywords
signed measures, polynomial certificates, growing negative index, Christoffel functions, tail scales, local concentration
Manuscript date
2026-10-25
Reader reuse
No additional permission granted

Scope

Sharp growing-index minimax degree for arbitrary shrinking negative probabilities under a containing-interval gap and distinct Gaussian or full exponential moment budgets. Separate single-atom gapless classes have matching local-concentration orders under uniformly superlinear or fixed nonnegative logarithmically strengthened exponential tails.

Disclosures

ChatGPT Sol 6.1 supported argument development, literature discovery, proof checking and drafting. Scott Schubr is responsible for the mathematical content and attribution. The research was self-funded. The author owns Luminary Oddities and its Research Archive.

Reproduction

20-page searchable manuscript PDF and matching editable LaTeX sources are provided. README.md supplies build instructions. Finite mathematical verification programs accompany the source package.

References and related resources

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  7. S. Schubr, Sharp Degree Bounds for Signed-Measure Certificates, 2026. Permanent Archive link (https://research.luminaryoddities.com/sharp-degree-bounds-signed-measure-certificates/v/1).
  8. S. Schubr, Tail Decay and Collision Barriers for Polynomial Certificates, 2026. Permanent Archive link (https://research.luminaryoddities.com/tail-decay-collision-barriers-polynomial-certificates/v/1).
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