bibkey: simon2008christoffel authors: Barry Simon year: 2008 title: “The Christoffel-Darboux Kernel” doi: null url: https://arxiv.org/abs/0806.1528v1 claim: “The Christoffel variational principle bounds a point mass using finite moments; for compact real support its limit recovers the point mass.” strata_touched: [] license: citation-only triage: anchor

Endpoint mass from finite moment observations

Source: https://arxiv.org/abs/0806.1528v1. Theorem 2.1, PDF p. 5, identifies the inverse Gram moment matrix with the polynomial reproducing-kernel coefficient matrix. Theorem 9.2, PDF p. 15, equation (9.6), gives the minimum of $\int |P{|}^{2}d\mu$ under $P(x)=1$, $\mathrm{deg}P\le d$, as $1/K_d(x,x)$. Theorem 9.7, PDF p. 17, equation (9.21), gives $K_d(x,x)\to \mu (\{x\}{)}^{-1}$ for compact real support, with infinity when the atom has mass zero.

The consumer is LocalClock, §十四, 定理27: $x=1$, support $[0,1]$, $H_d=[c_{i+j}]$ and $v=(1,\dots ,1{)}^T$. For $H_d$ positive definite this yields $B_d=(v^T{H}_d^{-1}v{)}^{-1}$. Singular matrices use the original constrained quadratic minimum; no inverse is asserted. The consumer also gives the elementary endpoint squeeze using $P(\lambda )={\lambda }^d$, so its finite-support cases do not require an infinite orthogonal polynomial family. This is a classical estimation certificate, not a new moment theorem or a finite-sample guarantee.