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  • Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev

Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/Extremal.lean

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@@ -19,18 +19,19 @@ following proof in https://math.stackexchange.com/a/978145/1277
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## Main statements
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* leadingCoeff_le_of_bounded: If `P` is a degree `n` real polynomial and `|P (x)| ≤ 1` for all `|x| ≤ 1` then
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the leading coefficient of `P` is at most `2 ^ (n-1)`
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* `leadingCoeff_eq_iff_of_bounded`: If P is a degree n polynomial and |P(x)|≤1 for all |x|≤1 then
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the leading coefficient of P equals 2^(n-1) iff P = T_n, the n'th Chebyshev polynomial
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* leadingCoeff_le_of_bounded: If `P` is a degree `n` real polynomial and `|P (x)| ≤ 1` for all
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`|x| ≤ 1` then the leading coefficient of `P` is at most `2 ^ (n-1)`
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* leadingCoeff_eq_iff_of_bounded: If `P` is a degree `n` polynomial and `|P (x)| ≤ 1` for all
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`|x| ≤ 1` then the leading coefficient of `P` equals `2 ^ (n-1)` iff `P = T_n`, the `n`'th
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Chebyshev polynomial
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-/
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@[expose] public section
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namespace Polynomial.Chebyshev
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open Polynomial Real
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/-- For n ≠ 0 and i ≤ n, `chebyshevNode n i` is one of the extremal points of the Chebyhsev T
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polynomial over the interval [-1, 1]. -/
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/-- For `n ≠ 0` and `i ≤ n`, chebyshevNode n i is one of the extremal points of the Chebyhsev T
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polynomial over the interval `[-1, 1]`. -/
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noncomputable abbrev chebyshevNode (n i : ℕ) : ℝ := cos (i * π / n)
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lemma chebyshevNode_eq_one {n : ℕ} : chebyshevNode n 0 = 1 := by simp [chebyshevNode]

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