LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Christoffel sampling is normalized and its weighted Gram has mean identity

Statement

Let $Λ$ and $K_{Λ}$ be as in Christoffel-weighted least-squares polynomial collocation, with $N = ∣ Λ ∣ \geq 1$ . Then $K_{Λ} \geq 1$ pointwise, the measure $d ν = (K_{Λ} / N) d ρ$ is a probability measure, and the random matrix $A (X) = \frac{N}{K_{Λ} (X)} ψ (X) ψ (X)^{T}, X \sim ν,$ satisfies $A ⪰ 0$ , $E_{ν} A = I_{N}$ , and $∥ A ∥_{o p} \leq N$ almost surely.

Facts & Assumptions

Given: A finite real orthonormal polynomial family indexed by $Λ$ , with $0 \in Λ$ , and its Christoffel kernel.

[F1]

Christoffel-weighted least-squares polynomial collocation defines $K_{Λ} = \sum_{α \in Λ} ∣ ψ_{α} ∣^{2}$ and the rank-one matrix $A$ .

[F2]

Independent polynomial-chaos tensor products are orthonormal gives $\int ψ_{α} ψ_{β} d ρ = δ_{α β}$ and $ψ_{0} = 1$ .

[F3]

Integration is linear on integrable functions (The Lebesgue integral is linear on $L^{1} (μ)$ ); the finite sums defining $K_{Λ}$ can therefore be integrated term by term.

[F4]

Cauchy-Schwarz inequality for $L^{2}$ makes each product $ψ_{α} ψ_{β}$ integrable. Since $d ν = (K_{Λ} / N) d ρ$ , integration against $ν$ is integration against that density for nonnegative measurable functions, and for signed functions once absolute integrability is established.

Proof

1.1F1F2F3algebra

Since $0 \in Λ$ and $ψ_{0} = 1$ , $K_{Λ} (y) \geq 1$ . By [F2]–[F3], $\int K_{Λ} d ρ = \sum_{α \in Λ} \int ∣ ψ_{α} ∣^{2} d ρ = N$ . Therefore $d ν = (K_{Λ} / N) d ρ$ is a probability measure and the denominator in [F1] is positive everywhere.

2.1F1F2F4step 1.1algebra∎

For each pair $α, β$ , [F4] and $\int ∣ ψ_{α} ψ_{β} ∣ d ρ \leq 1$ show $\int ∣ A_{α β} ∣ d ν = \int ∣ ψ_{α} ψ_{β} ∣ d ρ \leq 1$ , so the entry is integrable under $ν$ . Using the signed density identity in [F4], its expectation is $\int_{Γ} ψ_{α} (y) ψ_{β} (y) d ρ (y) = δ_{α β}$ by [F2], hence $E_{ν} A = I_{N}$ . The rank-one outer product is positive semidefinite and has operator norm $N ∥ ψ (y) ∥_{2}^{2} / K_{Λ} (y) = N$ , proving the remaining assertions.

Depends on

Used by

Bounded-Christoffel sampling stabilizes the empirical Gram matrix Theorem

Cited to discharge well-definedness by Christoffel-weighted least-squares polynomial collocation.

Dependency tree · two levels

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Sources

Albert Cohen and Giovanni Migliorati, Optimal Weighted Least-Squares Methods, §2 (standard reference, not scraped)
Joel A. Tropp, An Introduction to Matrix Concentration Inequalities, §6.2 (standard reference, not scraped)