DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Endpoint-preserving integrated Legendre projector

Definition

For $p \geq 1$ and $g \in L^{2} (- 1, 1)$ , define the degree- $(p - 1)$ Legendre orthogonal projection

Π_{p - 1} g : = \sum_{n = 0}^{p - 1} \frac{2 n + 1}{2} (\int_{- 1}^{1} g (t) P_{n} (t) d t) P_{n} .

For $c \in R$ , define the endpoint-anchored integrated projector

I_{p} (c, g) (x) : = c + \int_{- 1}^{x} Π_{p - 1} g (t) d t, - 1 \leq x \leq 1.

If an absolutely continuous function $f$ is represented as $f (x) = f (- 1) + \int_{- 1}^{x} f^{'} (t) d t$ with $f^{'} \in L^{2}$ , write $I_{p} f : = I_{p} (f (- 1), f^{'})$ . The finite sum and integral are defined for every such input; no choice axiom is used in this definition.

Depends on

Used by

Dependency tree · two levels

9 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

Joachim Schöberl, Numerical Methods for Partial Differential Equations, p-version FEM (standard reference, not scraped)
Leszek F. Demkowicz, Mathematical Theory of Finite Elements (standard reference, not scraped)