DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $f : [0, 1] \to R$ and $n \in N$ . Its Bernstein polynomial with index $n$ is

$B_{n} (f) (x) : = \sum_{k < n + 1} ι ⁣ (\binom{n}{k}) f ⁣ (\frac{ι (k)}{ι (n)}) x^{k} (1 - x)^{n - k} (0 \leq x \leq 1),$

when $n \geq 1$ . 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 $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ . 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 · two levels

23 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

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

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