Bernstein approximations and Weierstrass' theorem

We prove that the Bernstein approximations

∑ k : Fin (n+1), (n.choose k * x^k * (1-x)^(n-k)) • f (k/n : ℝ)

for a continuous function f : C([0,1], E) taking values in a locally convex vector space converge uniformly to f as n tends to infinity. This statement directly applies to the cases when the codomain is a (semi)normed space or, more generally, has a topology defined by a family of seminorms.

Our proof follows [Richard Beals' Analysis, an introduction][beals-analysis], §7D. The original proof, due to Bernstein in 1912, is probabilistic, and relies on Bernoulli's theorem, which gives bounds for how quickly the observed frequencies in a Bernoulli trial approach the underlying probability.

The proof here does not directly rely on Bernoulli's theorem, but can also be given a probabilistic account.

• Consider a weighted coin which with probability x produces heads, and with probability 1-x produces tails.
• The value of bernstein n k x is the probability that such a coin gives exactly k heads in a sequence of n tosses.
• If such an appearance of k heads results in a payoff of f(k / n), the n-th Bernstein approximation for f evaluated at x is the expected payoff.
• The main estimate in the proof bounds the probability that the observed frequency of heads differs from x by more than some δ, obtaining a bound of (4 * n * δ^2)⁻¹, irrespective of x.
• This ensures that for n large, the Bernstein approximation is (uniformly) close to the payoff function f.

(You don't need to think in these terms to follow the proof below: it's a giant calc block!)

This result proves Weierstrass' theorem that polynomials are dense in C([0,1], ℝ), although we defer an abstract statement of this until later.

def bernstein (n ν : ℕ) : C(↑unitInterval, ℝ)

The Bernstein polynomials, as continuous functions on [0,1].

Equations

• bernstein n ν = (bernsteinPolynomial ℝ n ν).toContinuousMapOn unitInterval

theorem bernstein_apply (n ν : ℕ) (x : ↑unitInterval) : (bernstein n ν) x = ↑(n.choose ν) * ↑x ^ ν * (1 - ↑x) ^ (n - ν)

@[simp]

theorem bernstein_nonneg {n ν : ℕ} {x : ↑unitInterval} : 0 ≤ (bernstein n ν) x

def Mathlib.Meta.Positivity.evalBernstein : PositivityExt

Extension of the positivity tactic for Bernstein polynomials: they are always non-negative.

Equations

• One or more equations did not get rendered due to their size.

We now give a slight reformulation of bernsteinPolynomial.variance.

def bernstein.z {n : ℕ} (k : Fin (n + 1)) : ↑unitInterval

Send k : Fin (n+1) to the equally spaced points k/n in the unit interval.

Equations

• bernstein.z k = ⟨↑↑k / ↑n, ⋯⟩

@[simp]

theorem bernstein.z_zero {n : ℕ} : z 0 = 0

@[simp]

theorem bernstein.z_last {n : ℕ} (hn : n ≠ 0) : z (Fin.last n) = 1

@[simp]

theorem bernstein.probability (n : ℕ) (x : ↑unitInterval) : ∑ k : Fin (n + 1), (bernstein n ↑k) x = 1

theorem bernstein.variance {n : ℕ} (hn : n ≠ 0) (x : ↑unitInterval) : ∑ k : Fin (n + 1), (↑x - ↑(z k)) ^ 2 * (bernstein n ↑k) x = ↑x * (1 - ↑x) / ↑n

def bernsteinApproximation {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] (n : ℕ) (f : C(↑unitInterval, E)) : C(↑unitInterval, E)

The n-th approximation of a continuous function on [0,1] by Bernstein polynomials, given by ∑ k, bernstein n k x • f (k/n).

Equations

• bernsteinApproximation n f = ∑ k : Fin (n + 1), bernstein n ↑k • ContinuousMap.const (↑unitInterval) (f (bernstein.z k))

We now set up some of the basic machinery of the proof that the Bernstein approximations converge uniformly.

A key player is the set S f ε h n x, for some function f : C(I, E), h : 0 < ε, n : ℕ and x : I.

This is the set of points k in Fin (n+1) such that k/n is within δ of x, where δ is the modulus of uniform continuity for f, chosen so ‖f x - f y‖ < ε/2 when |x - y| < δ.

We show that if k ∉ S, then 1 ≤ δ^-2 * (x - k/n)^2.

theorem bernsteinApproximation.apply {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] (n : ℕ) (f : C(↑unitInterval, E)) (x : ↑unitInterval) : (bernsteinApproximation n f) x = ∑ k : Fin (n + 1), (bernstein n ↑k) x • f (bernstein.z k)

@[simp]

theorem bernsteinApproximation.apply_zero {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] (n : ℕ) (f : C(↑unitInterval, E)) : (bernsteinApproximation n f) 0 = f 0

@[simp]

theorem bernsteinApproximation.apply_one {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] {n : ℕ} (hn : n ≠ 0) (f : C(↑unitInterval, E)) : (bernsteinApproximation n f) 1 = f 1

theorem bernsteinApproximation_uniform {E : Type u_1} [AddCommGroup E] [TopologicalSpace E] [IsTopologicalAddGroup E] [Module ℝ E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (f : C(↑unitInterval, E)) : Filter.Tendsto (fun (n : ℕ) => bernsteinApproximation n f) Filter.atTop (nhds f)

The Bernstein approximations

∑ k : Fin (n+1), f (k/n : ℝ) * n.choose k * x^k * (1-x)^(n-k)

for a continuous function f : C([0,1], ℝ) converge uniformly to f as n tends to infinity.

This is the proof given in [Richard Beals' Analysis, an introduction][beals-analysis], §7D, and reproduced on wikipedia.