All Research

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.