| import OAI.Geometry.RCD.Metric.QuantitativeDimensionalCross |
|
|
| namespace OAI |
|
|
| /-! # A quantitative frame-energy identity from centered square data -/ |
|
|
| open Finset Real |
| noncomputable section |
| namespace RegularPointCoordinates |
|
|
| theorem bochner_frame_defect {E : Type*} [AddCommGroup E] [Module ℝ E] |
| {n : ℕ} (hn : 0<n) (B : E →ₗ[ℝ] E →ₗ[ℝ] ℝ) (L : E →ₗ[ℝ] ℝ) |
| (hsym : ∀ a b, B a b=B b a) (hBE : ∀ a, (L a)^2≤(n:ℝ)*B a a) |
| (z : Fin n → E) (w : E) (G : Fin n → Fin n → ℝ) |
| (r : ℝ) (a : Fin n → ℝ) {D : ℝ} (hD : 0≤D) (hD1 : D≤1) |
| (hLz : ∀ i, L (z i)=2*G i i) |
| (hBz : ∀ i j, B (z i) (z j)=4*(G i j)^2) |
| (hLw : L w=2*r) (hBw : B w w=4*r^2) |
| (hcross : ∀ i, B (z i) w=4*(a i)^2) |
| (htrace : |∑ i, G i i-(n:ℝ)|≤D) |
| (hsquare : |∑ i, ∑ j, (G i j)^2-(n:ℝ)|≤D) : |
| |r-∑ i, (a i)^2|≤√(6*D)*|r| := by |
| classical |
| let q := (1/2:ℝ) • ∑ i, z i |
| let v := (1/2:ℝ) • w |
| have hLq : L q=∑ i, G i i := by |
| simp only [q,map_smul,map_sum,hLz,smul_eq_mul,← mul_sum] |
| ring |
| have hLv : L v=r := by simp [v,hLw] |
| have hBq : B q q=∑ i, ∑ j, (G i j)^2 := by |
| conv_rhs => rw [Finset.sum_comm] |
| simp only [q,map_smul,map_sum,LinearMap.smul_apply,LinearMap.sum_apply,smul_eq_mul,hBz, |
| ← mul_sum] |
| ring |
| have hBv : B v v=r^2 := by simp [v,hBw]; ring |
| have hqv : B q v=∑ i, (a i)^2 := by |
| simp only [q,v,map_smul,map_sum,LinearMap.smul_apply,LinearMap.sum_apply,smul_eq_mul, |
| hcross,← mul_sum] |
| ring |
| have hvq : B v q=∑ i, (a i)^2 := (hsym v q).trans hqv |
| apply dimensional_cross_abs_defect (N := (n:ℝ)) (by exact_mod_cast hn) |
| hD hD1 htrace hsquare |
| intro t |
| have hh := hBE (q+t • v) |
| simp only [map_add,map_smul,LinearMap.add_apply,LinearMap.smul_apply,smul_eq_mul, |
| hLq,hLv,hBq,hBv,hqv,hvq] at hh |
| convert hh using 1 |
| ring |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
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| end OAI |
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|
