LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The cellwise L2 projection exists uniquely and is best approximation

Statement

For every finite measure $μ$ on a bounded nondegenerate interval or nondegenerate affine simplex $K$ that assigns positive measure to every nonempty relatively open subset, every integer degree $p \geq 0$ with $P_{p} (K) \subseteq L^{2} (μ)$ as in Cellwise L2 projection relation onto broken polynomials, and every $u \in L^{2} (μ)$ , the set $P_{K} (u)$ contains exactly one polynomial $Π_{K} u$ . For each $w \in P_{p} (K)$ , $∥ u - Π_{K} u ∥_{L^{2} (K)} \leq ∥ u - w ∥_{L^{2} (K)} .$

Facts & Assumptions

Given: A finite measure $μ$ with full support on $K$ , the finite-dimensional polynomial space $P_{p} (K) \subset L^{2} (μ)$ , $u \in L^{2} (μ)$ , and the orthogonality equations defining $P_{K} (u)$ .

[F1]

Cellwise L2 projection relation onto broken polynomials: $p \in P_{K} (u)$ exactly when $\int_{K} (u - p) w d μ = 0$ for every $w \in P_{p} (K)$ .

[F2]

Cauchy-Schwarz inequality for $L^{2}$ : if $f, g \in L^{2} (μ)$ , then $\int ∣ f g ∣ d μ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ .

Proof

1.1F1F2constructalgebra

Choose the finite monomial basis ${ϕ_{1}, \dots, ϕ_{M}}$ of $P_{p} (K)$ . Its Gram matrix $G_{i j} = \int_{K} ϕ_{i} ϕ_{j} d μ$ is positive definite: for every nonzero coefficient vector $c$ , the polynomial $q = \sum_{i} c_{i} ϕ_{i}$ is continuous and nonzero at some point of $K$ , so it is bounded away from zero on a nonempty relatively open subset; full support gives $c^{T} G c = \int_{K} q^{2} d μ > 0$ . Each moment $b_{i} = \int_{K} u ϕ_{i} d μ$ is finite by [F2]. The invertible system $G c = b$ therefore has a unique solution, and $p = \sum_{i} c_{i} ϕ_{i}$ satisfies [F1].

2.1F1step 1.1algebra∎

For any $w \in P_{p} (K)$ , write $u - w = (u - Π_{K} u) + (Π_{K} u - w)$ . The first term is orthogonal to the second by [F1], hence $∥ u - w ∥_{2}^{2} = ∥ u - Π_{K} u ∥_{2}^{2} + ∥ Π_{K} u - w ∥_{2}^{2} \geq ∥ u - Π_{K} u ∥_{2}^{2}$ . Taking square roots proves best approximation.

Depends on

Used by

Smooth-solution L2 error bound for upwind DG transport Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bernardo Cockburn, An Introduction to the Discontinuous Galerkin Method for Convection-Dominated Problems, CIME lecture notes (standard reference, not scraped)