OpenFrontiers
Join
← Back to the index
OP-6960Status unassessedAnalysis

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

TheoremDB P9854 · Candidate awaiting source review

Source candidate · not reviewedAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Fix a real number α\alpha with α>0\alpha > 0 and α≠1\alpha \neq 1. For each n∈Nn \in \mathbb{N}, the Bernstein basis polynomial of degree nn and index jj is

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

For 0≤k≤n0 \leq k \leq n, define the cumulative sum

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

and set Jn,n+1(x)=0J_{n,n+1}(x) = 0. The Bézier-type Bernstein operator Bn,αB_{n,\alpha} acts on a function f:[0,1]→Rf: [0,1] \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 whether there exists a function L:(R→R)×[0,1]→RL: (\mathbb{R} \to \mathbb{R}) \times [0,1] \to \mathbb{R} and an integer m0∈Nm_0 \in \mathbb{N} such that for every integer m≥m0m \geq m_0, every function f:R→Rf: \mathbb{R} \to \mathbb{R} that is CmC^m on [0,1][0,1], and every x∈[0,1]x \in [0,1], the following asymptotic formula holds:

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

If such a limit formula exists, find an explicit expression for L(f,x)L(f,x) in terms of ff, xx, and α\alpha.

Lean 4

No Lean submitted

Formal statement

Lean 4
theorem voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth
    (α : ℝ) (hα_pos : 0 < α) (hα : α ≠ 1) :
    let limitFormula : (ℝ → ℝ) → ℝ → ℝ := answer(sorry)
    ∀ᶠ m : ℕ in atTop,
      ∀ (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))

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.

Proofs & verification

No Lean has been submitted for this problem yet. A Lean file can be attached to a submission for review, though with no target here it cannot be machine-checked.

Share a formalization →Verification guide ↗

Progress & source assessment

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

Open. No signed Lean verification is attached to this problem or its selected resolution.

This record is in the source backlog; its novelty and unresolved status have not passed source review.

References
  1. [1]TheoremDB P9854 link
  2. [2]Source formal statement (not locally verified) link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.