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OP-6959Status unassessedAnalysis

Voronovskaja-type asymptotic formula for Bézier–Bernstein operators with shape parameter

TheoremDB P8636 · Candidate awaiting source review

Source candidate · not reviewedAs of 2026-09-08Source & evidence ↓
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Problem statement

Let I=[0,1]I = [0,1] and fix a real parameter α>0\alpha > 0 with α≠1\alpha \neq 1. For each n∈Nn \in \mathbb{N}, the Bernstein basis polynomial is defined by

pn,j(x)=(nj)xj(1−x)n−j p_{n,j}(x) = \binom{n}{j} x^j (1-x)^{n-j}

for j=0,1,…,nj = 0, 1, \ldots, n, and the cumulative tail sum is

Jn,k(x)=∑j=knpn,j(x) J_{n,k}(x) = \sum_{j=k}^{n} p_{n,j}(x)

for k=0,1,…,nk = 0, 1, \ldots, n, with Jn,n+1(x)=0J_{n,n+1}(x) = 0. The Bézier-type Bernstein operator Bn,αB_{n,\alpha} acts on a function f:I→Rf: I \to \mathbb{R} by

(Bn,αf)(x)=∑k=0nf ⁣(kn)(Jn,k(x)α−Jn,k+1(x)α). (B_{n,\alpha} f)(x) = \sum_{k=0}^{n} f\!\left(\frac{k}{n}\right) \Bigl( J_{n,k}(x)^{\alpha} - J_{n,k+1}(x)^{\alpha} \Bigr).

Determine the smallest natural number mm and an explicit formula Λ(f,x)\Lambda(f,x) such that for every function f:R→Rf: \mathbb{R} \to \mathbb{R} and every point x∈Ix \in I, if ff is CmC^m on II (that is, mm-times continuously differentiable on II with all derivatives up to order mm continuous on II), then

lim⁡n→∞n ((Bn,αf)(x)−f(x))=Λ(f,x). \lim_{n \to \infty} \sqrt{n}\, \bigl( (B_{n,\alpha} f)(x) - f(x) \bigr) = \Lambda(f,x).

Equivalently, determine the pair (m,Λ)(m, \Lambda) where m∈Nm \in \mathbb{N} and Λ:(R→R)×I→R\Lambda: (\mathbb{R} \to \mathbb{R}) \times I \to \mathbb{R} is a functional, such that the sequence n(Bn,αf(x)−f(x))\sqrt{n}\bigl( B_{n,\alpha} f(x) - f(x) \bigr) converges to Λ(f,x)\Lambda(f,x) for all CmC^m functions ff and all x∈Ix \in I.

Lean 4

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Formal statement

Lean 4
theorem voronovskaja_theorem.bezier_bernstein_operators.variants.answer_smoothness
    (α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :
    let p : ℕ × ((ℝ → ℝ) → ℝ → ℝ) := answer(sorry)
    let m := p.1
    let limitFormula := p.2
    ∀ (f : ℝ → ℝ) (x : ℝ), x ∈ I → ContDiffOn ℝ m f I →
      Tendsto (fun n : ℕ => Real.sqrt n * (bezierBernstein n α f x - f x)) atTop
        (𝓝 (limitFormula f x))

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Source-reported assessment, retrieved 8 September 2026. Frontier Science has not independently reviewed this status.

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References
  1. [1]TheoremDB P8636 link
  2. [2]Source formal statement (not locally verified) link

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