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.

Natural cubic splines

Definition

Let $[a, b]$ be a bounded interval and let $a \leq t_{1} < \dots < t_{r} \leq b$ with $r \geq 2$ . A natural cubic spline with knots $t_{1}, \dots, t_{r}$ is a $C^{2}$ function on $[a, b]$ that is a polynomial of degree at most three on every $[t_{j}, t_{j + 1}]$ , has second derivative zero at $t_{1}$ and $t_{r}$ , and is extended linearly on $[a, t_{1}]$ and $[t_{r}, b]$ when those intervals are nonempty. The knots $t_{j}$ are the distinct breakpoints; repeated observations at the same input do not create additional knots.

Used by

Clamped cubic spline interpolant Definition
Natural cubic smoothing spline Definition
A natural cubic spline undershoots a nonnegative table Example
Natural cubic interpolation with two and three knots Example
The natural interpolant of x squared has a second-order endpoint error Example
Nodal second-derivative perturbations bound the natural spline error Lemma
Existence, uniqueness, and spline reduction for cubic smoothing Theorem
Natural cubic interpolation at distinct knots Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ryan Tibshirani, Nonparametric Regression: Splines and RKHS Methods, §§2.1–2.3 (standard reference, not scraped)