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Growing-Index Recovery and Tail Scales for Signed Measures

Scott Schubr

  • Mathematics
  • Classical and Real Analysis
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

We study uniform recovery of negative directions by polynomial squares when the requested negative index and the tail resource vary. For a shrinking negative probability measure κ, write h_m for its degree-m monic squared norm and L=log(R/h_m). Under a gap from its containing interval, we prove that the minimax degree for recovering m+1 negative directions is asymptotic to L^2/(8cd^2) in a Gaussian background class. In a separately specified full exponential-moment class, its logarithm is asymptotic to π L/(2cd). No additional restriction on the growth of the index is needed: the negative measure may be atomic, continuous, singular or mixed, and its support concentration pays the dimension costs. Adaptive rational damping and a subleading Fourier extension retain the sharp constants. Without a support gap, a local mass bound A_0Rr^q and tails exp(c|x|[log(e+|x|)]^β) give log D=Θ(R^(1/[q(β+1)])) for every fixed β≥0 and q>0. For a uniformly superlinear tail scale H, we obtain D=Θ(F^(-1)(R^(1/q))), where F(t)=t/H^(-1)(t), including every pure power greater than one. The proofs distinguish these signed-measure recovery conclusions from their classical orthogonal-polynomial and localization ingredients.

Keywords signed measures · polynomial certificates · growing negative index · Christoffel functions · tail scales · local concentration

Scope of the work

Sharp growing-index signed-measure polynomial recovery laws under specified Gaussian, exponential and superlinear tail-resource classes, with arbitrary negative-support type and explicit gap/local-mass hypotheses.

Reproduction notes

19-page manuscript and 10 source-package members. Compile main.tex with its included section and reference files. Standard-library Python runs python verify_finite_controls.py, also under -O, for the exact finite Gram and recovery controls accompanying the asymptotic arguments.

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Author disclosures

AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author declares no competing interests.

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Cite this edition

Scott Schubr. (2026). Growing-Index Recovery and Tail Scales for Signed Measures (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/growing-index-recovery-tail-scales-signed-measures/v/2

BibTeX
@techreport{luminary_growing_index_recovery_tail_scales_signed_measures_v2,
  title = {Growing-Index Recovery and Tail Scales for Signed Measures},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/growing-index-recovery-tail-scales-signed-measures/v/2}
}

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