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cead56f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | 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
end OAI
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