LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

Polynomial coefficients are bounded by values at fixed distinct nodes

Statement

Let $n \geq 0$ and let $t_{0}, \dots, t_{n}$ be distinct real numbers. For $0 \leq j \leq n$ , define $L_{j} (x) : = \prod_{k \neq j} \frac{x - t_{k}}{t_{j} - t_{k}} .$ If $p (x) = \sum_{k = 0}^{n} c_{k} x^{k}$ has degree at most $n$ , then $p (x) = \sum_{j = 0}^{n} p (t_{j}) L_{j} (x) .$ Consequently, writing $L_{j} (x) = \sum_{k = 0}^{n} ℓ_{j k} x^{k}$ , every coefficient satisfies $∣ c_{k} ∣ \leq (\sum_{j = 0}^{n} ∣ ℓ_{j k} ∣) \max_{0 \leq j \leq n} ∣ p (t_{j}) ∣ .$ The empty product convention gives $L_{0} = 1$ when $n = 0$ .

Facts & Assumptions

Given: An integer $n \geq 0$ , distinct real nodes $t_{0}, \dots, t_{n}$ , and a real polynomial $p$ of degree at most $n$ .

[F1]

Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials: a polynomial of degree at most $n$ is a finite coefficient sum $\sum_{k = 0}^{n} c_{k} x^{k}$ .

[F2]

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots: a nonzero polynomial of degree at most $n$ has at most $n$ distinct real roots.

Proof

1.1F1F2givenconstructalgebra

Each denominator $t_{j} - t_{k}$ is nonzero, so every $L_{j}$ is a polynomial of degree at most $n$ and $L_{j} (t_{i})$ is $1$ for $i = j$ and $0$ otherwise. The polynomial $p - \sum_{j} p (t_{j}) L_{j}$ has degree at most $n$ and vanishes at the $n + 1$ distinct nodes; [F2] makes it the zero polynomial. This also covers $n = 0$ , when $L_{0} = 1$ .

2.1givenstep 1.1algebra∎

Comparing the coefficients of $x^{k}$ in the identity of step 1.1 gives $c_{k} = \sum_{j} p (t_{j}) ℓ_{j k}$ . The triangle inequality yields the stated coefficient bound.

Depends on

Used by

Best uniform polynomial approximants exist Theorem
Best weighted rational approximants exist Theorem

Dependency tree · two levels

6 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

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Polynomial interpolation (standard reference, not scraped)
Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chapter 5 (standard reference, not scraped)