feat: Voronovskaja-type formula for the Bézier variant of the Bernstein operators

FormalConjectures/Paper/VoronovskajaTypeFormula.lean

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+/-
+Copyright 2025 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+
+open Topology Filter Real unitInterval Polynomial
+
+/-!
+# Voronovskaja-type Formula for the Bezier Variant of the Bernstein Operators
+
+*References:*
+
+* [A problem in Constructive theory of functions, Szopol 2010](https://www.math.bas.bg/mathmod/Proceedings_CTF/CTF-2010/files_CTF-2010/Open_problems.pdf?utm_source=perplexity)
+
+The Bézier-type Bernstein operators $B_{n,\alpha}$ for $\alpha > 0$ are defined for
+$f : [0,1] \to \mathbb{R}$ by
+\[
+(B_{n,\alpha} f)(x)
+  = \sum_{k=0}^n f\!\left(\frac{k}{n}\right)
+    \left( J_{n,k}(x)^{\alpha} - J_{n,k+1}(x)^{\alpha} \right),
+\]
+where
+\[
+J_{n,k}(x) = \sum_{j=k}^n p_{n,j}(x),
+\qquad
+p_{n,j}(x) = \binom{n}{j} x^j(1-x)^{n-j},
+\]
+and $J_{n,n+1}(x) = 0$.
+
+In the classical case $\alpha = 1$, these operators reduce to the usual Bernstein operators.
+For sufficiently smooth $f$, one has the classical Voronovskaja asymptotic formula
+\[
+\lim_{n \to \infty} n\bigl( B_{n,1} f(x) - f(x) \bigr)
+    = \tfrac{1}{2} x(1-x) f''(x).
+\]
+
+## Known Results
+* For $\alpha = 1$, the asymptotics are completely understood.
+* Numerical experiments indicate that for $\alpha \neq 1$ the quantity
+    \[
+        \sqrt{n}\,\bigl( B_{n,\alpha} f(x) - f(x) \bigr)
+    \]
+    may converge to a non-zero limit.
+
+## The Problem
+Determine the asymptotic behaviour of the Bézier-type Bernstein operators for $\alpha \neq 1$:
+\textbf{Existence of the limit:}
+    Prove (or disprove) the existence of the limit
+    \[
+        \lim_{n \to \infty}
+        \sqrt{n}\,\bigl( B_{n,\alpha} f(x) - f(x) \bigr),
+    \]
+    at least for functions $f$ that are twice differentiable at $x \in (0,1)$.
+    \item \textbf{Explicit form of the limit:}
+    If the limit exists, determine an explicit expression for it in terms of $f$, $x$, and $\alpha$.
+-/
+
+/--
+Cumulative sum `J_{n,k}(x) = ∑_{j=k}^n p_{n,j}(x)`
+-/
+noncomputable def BernsteinTail (n k : ℕ) : Polynomial ℝ :=
+  ∑ j ∈ Finset.Icc k n,
+    bernsteinPolynomial ℝ n j
+
+/--
+Bézier–type Bernstein operator:
+`(B_{n,α} f)(x) = ∑_{k=0}^n f(k/n) * (J_{n,k}(x)^α - J_{n,k+1}(x)^α)`
+-/
+noncomputable def BezierBernstein (n : ℕ) (α : ℝ) (f : ℝ  → ℝ) (x : ℝ) : ℝ :=
+∑ k ∈  Finset.range (n+1),
+    f (k/n) * (((BernsteinTail n k).eval x) ^ α - ((BernsteinTail n k + 1).eval x) ^ α)
+
+/--
+Classical Voronovskaja theorem (α = 1)
+
+For smooth `f`, the limit:
+    n * (B_n f x - f x) → (1/2)*x*(1-x)*f''(x)
+
+This is already in the literature; here we state it.
+-/
+@[category research solved, AMS 26 40 47]
+theorem voronovskaja_theorem.bernstein_operators
+    (f : ℝ → ℝ) (x : ℝ) (hx : x ∈ I)
+    (f'' : ℝ := iteratedDerivWithin 2 f I x) :
+    Tendsto (fun (n : ℕ) => n • (BezierBernstein n 1 f x - f x))
+    atTop
+    (𝓝 ((1/2) * x * (1-x) * f'')) := by
+  sorry
+
+/--
+Conjecture: Voronovskaja-type formula for Bézier-Bernstein operators
+with shape parameter α ≠ 1.
+-/
+@[category research open, AMS 26 40 47]
+theorem voronovskaja_theorem.bezier_bernstein_operators
+    (α : ℝ) (hα : α ≠ 1)
+    (f : ℝ → ℝ) (x : ℝ) (hx : x ∈ I)
+    (f'' : ℝ := iteratedDerivWithin 2 f I x)
+    (x : ℝ)(hf : ContDiffAt ℝ 2 f x) : ∃ L : ℝ,
+    Tendsto (fun n : ℕ => Real.sqrt n * (BezierBernstein n α f x - f x)) atTop (𝓝 L) := by
+  sorry