| import Mathlib.Analysis.InnerProductSpace.Adjoint |
| import Mathlib.Analysis.InnerProductSpace.Projection.Submodule |
| import Mathlib.Analysis.SpecificLimits.Basic |
| import Mathlib.Algebra.Ring.GeomSum |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Abel |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology BigOperators |
| namespace Nonhyperlinear.Diffuseness |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| / |
| def cesaro (S : H →L[ℂ] H) (n : ℕ) : H →L[ℂ] H := |
| ((n+1 : ℕ) : ℂ)⁻¹ • ∑ k ∈ Finset.range (n+1), S^k |
|
|
| omit [CompleteSpace H] in |
| lemma pow_norm_le_one [CompleteSpace H] (S : H →L[ℂ] H) (hS : ‖S‖≤1) (k : ℕ) : ‖S^k‖≤1 := |
| by |
| induction k with |
| | zero => |
| rw [pow_zero] |
| apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one |
| intro x |
| simp |
| | succ k ih => |
| rw [pow_succ] |
| exact (norm_mul_le _ _).trans (by nlinarith [norm_nonneg (S^k),norm_nonneg S]) |
|
|
| lemma cesaro_norm_le_one (S : H →L[ℂ] H) (hS : ‖S‖≤1) (n : ℕ) : ‖cesaro S n‖≤1 := by |
| have hn : (0:ℝ)<(n+1:ℕ) := Nat.cast_pos.mpr (Nat.succ_pos n) |
| have hs : ‖∑ k ∈ Finset.range (n+1), S^k‖≤(n+1:ℕ) := by |
| calc |
| _ ≤ ∑ k ∈ Finset.range (n+1), ‖S^k‖ := norm_sum_le _ _ |
| _ ≤ ∑ k ∈ Finset.range (n+1), (1:ℝ) := Finset.sum_le_sum (fun k _ => pow_norm_le_one S hS k) |
| _ = _ := by simp |
| rw [cesaro,norm_smul,norm_inv] |
| simp only [Complex.norm_natCast] |
| exact (mul_le_mul_of_nonneg_left hs (inv_nonneg.mpr hn.le)).trans_eq (inv_mul_cancel₀ hn.ne') |
|
|
| lemma cesaro_apply_norm_le (S : H →L[ℂ] H) (hS : ‖S‖≤1) (n : ℕ) (x : H) : |
| ‖cesaro S n x‖≤‖x‖ := |
| ((cesaro S n).le_opNorm x).trans (by nlinarith [cesaro_norm_le_one S hS n,norm_nonneg x]) |
|
|
| omit [CompleteSpace H] in |
| lemma cesaro_mul_sub [CompleteSpace H] (S : H →L[ℂ] H) (n : ℕ) : |
| cesaro S n*(1-S)=((n+1:ℕ):ℂ)⁻¹ • (1-S^(n+1)) := by |
| rw [cesaro,smul_mul_assoc,geom_sum_mul_neg] |
|
|
| lemma cesaro_range_bound (S : H →L[ℂ] H) (hS : ‖S‖≤1) (n : ℕ) (x : H) : |
| ‖cesaro S n ((1-S) x)‖≤((n+1:ℕ):ℝ)⁻¹*(2*‖x‖) := by |
| have hn : (0:ℝ)<(n+1:ℕ) := Nat.cast_pos.mpr (Nat.succ_pos n) |
| have hp : ‖(S^(n+1)) x‖≤‖x‖ := |
| ((S^(n+1)).le_opNorm x).trans (by nlinarith [pow_norm_le_one S hS (n+1),norm_nonneg x]) |
| have he := congrArg (fun T:H→L[ℂ]H => T x) (cesaro_mul_sub S n) |
| simp only [mul_apply_eq_comp,smul_apply] at he |
| rw [he,norm_smul,norm_inv] |
| simp only [Complex.norm_natCast] |
| apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hn.le) |
| change ‖x-(S^(n+1)) x‖≤_ |
| exact (norm_sub_le _ _).trans (by linarith) |
|
|
| lemma cesaro_on_range (S : H →L[ℂ] H) (hS : ‖S‖≤1) (x : H) : |
| Tendsto (fun n => cesaro S n ((1-S) x)) atTop (𝓝 0) := by |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| have ht := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ)).mul_const (2*‖x‖) |
| simp only [zero_mul,one_div] at ht |
| apply squeeze_zero (fun n=>norm_nonneg _) (cesaro_range_bound S hS · x) |
| simpa only [Nat.cast_add,Nat.cast_one] using ht |
|
|
| end Nonhyperlinear.Diffuseness |
|
|
| end |
|
|
| end OAI |
|
|
