Growing-Index Recovery and Tail Scales for Signed Measures

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 kappa, let h_m be its degree-m monic squared norm and L=log(R/h_m). Under a gap from its containing interval, 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 pi L/(2cd). No additional restriction on the growth of the index is needed: the negative measure may be atomic, continuous, singular or mixed, and 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_0 R r^q and tails exp(c|x|[log(e+|x|)]^beta) give log D=Theta(R^{1/[q(beta+1)]}) for every fixed beta>=0 and q>0. For a uniformly superlinear tail scale H, D=Theta(F^{-1}(R^{1/q})), where F(t)=t/H^{-1}(t), including every pure power greater than one. The proofs distinguish these recovery conclusions from their classical orthogonal-polynomial and localization ingredients.