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Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev Expand file tree Collapse file tree Original file line number Diff line number Diff line change @@ -19,18 +19,19 @@ following proof in https://math.stackexchange.com/a/978145/1277
1919
2020## Main statements
2121
22- * leadingCoeff_le_of_bounded: If `P` is a degree `n` real polynomial and `|P (x)| ≤ 1` for all `|x| ≤ 1` then
23- the leading coefficient of `P` is at most `2 ^ (n-1)`
24- * `leadingCoeff_eq_iff_of_bounded`: If P is a degree n polynomial and |P(x)|≤1 for all |x|≤1 then
25- the leading coefficient of P equals 2^(n-1) iff P = T_n, the n'th Chebyshev polynomial
22+ * leadingCoeff_le_of_bounded: If `P` is a degree `n` real polynomial and `|P (x)| ≤ 1` for all
23+ `|x| ≤ 1` then the leading coefficient of `P` is at most `2 ^ (n-1)`
24+ * leadingCoeff_eq_iff_of_bounded: If `P` is a degree `n` polynomial and `|P (x)| ≤ 1` for all
25+ `|x| ≤ 1` then the leading coefficient of `P` equals `2 ^ (n-1)` iff `P = T_n`, the `n`'th
26+ Chebyshev polynomial
2627 -/
2728@[expose] public section
2829namespace Polynomial.Chebyshev
2930
3031open Polynomial Real
3132
32- /-- For n ≠ 0 and i ≤ n, ` chebyshevNode n i` is one of the extremal points of the Chebyhsev T
33- polynomial over the interval [-1, 1]. -/
33+ /-- For ` n ≠ 0` and ` i ≤ n`, chebyshevNode n i is one of the extremal points of the Chebyhsev T
34+ polynomial over the interval ` [-1, 1]` . -/
3435noncomputable abbrev chebyshevNode (n i : ℕ) : ℝ := cos (i * π / n)
3536
3637lemma chebyshevNode_eq_one {n : ℕ} : chebyshevNode n 0 = 1 := by simp [chebyshevNode]
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