DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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 Bernstein polynomial $B_n(f)$ on $[0,1]$

Definition

Let $f:[0,1]\to\mathbb R$ and $n\in\mathbb N$ . Its Bernstein polynomial with index $n$ is

$B_n(f)(x):=\sum_{k<n+1}\iota\!\binom nk f\!\left(\frac{\iota(k)}{\iota(n)}\right)x^k(1-x)^{n-k}\qquad(0\le x\le1),$

when $n\ge1$ . It is a polynomial of degree at most $n$ when nonzero, and it may be the zero polynomial (for example when $f=0$ ). Here the finite sum and binomial coefficients are those of Finite sums and finite products, by recursion and The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$ . For $n=0$ set $B_0(f)(x):=f(0)$ ; this separate clause avoids the undefined quotient $0/0$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 59 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Bernstein polynomial (Encyclopedia of Mathematics) (standard reference, not scraped)

The Bernstein polynomial Bn(f)B_n(f)Bn​(f) on [0,1][0,1][0,1]