File size: 3,540 Bytes
59d1c80 | 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Geometry.IsometricImmersion.Assembly.FlatPatchSum
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Filter Set
namespace SmoothLocal.Geometry
variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V]
omit [CompleteSpace V] in
theorem exists_positive_diagonal_amplitudes (f : ℕ → Coord → V)
(hf : ∀ n, ContDiff ℝ ∞ (f n)) (hc : ∀ n, HasCompactSupport (f n))
(ε : ℝ) (hε : 0 < ε) :
∃ δ : ℕ → ℝ, (∀ n, 0 < δ n) ∧
∀ n k : ℕ, k ≤ n → ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun q => δ n • f n q) p‖ ≤ patchSeriesWeight ε n := by
classical
have hC (k n : ℕ) : ∃ C : ℝ, ∀ p, ‖iteratedFDeriv ℝ k (f n) p‖ ≤ C :=
compactSmooth_has_derivative_bounds (f n) (hf n) (hc n) k
choose C hC using hC
let D : ℕ → ℝ := fun n => 1 + ∑ k ∈ Finset.range (n + 1), |C k n|
have hD (n : ℕ) : 0 < D n := by
have hs : 0 ≤ ∑ k ∈ Finset.range (n + 1), |C k n| :=
Finset.sum_nonneg (fun k _ => abs_nonneg (C k n))
dsimp [D]
linarith
have hCD (n k : ℕ) (hkn : k ≤ n) : C k n ≤ D n := by
have hk : k ∈ Finset.range (n + 1) := Finset.mem_range.mpr (Nat.lt_succ_of_le hkn)
have hs : |C k n| ≤ ∑ j ∈ Finset.range (n + 1), |C j n| :=
Finset.single_le_sum (fun j _ => abs_nonneg (C j n)) hk
dsimp [D]
linarith [le_abs_self (C k n)]
let δ : ℕ → ℝ := fun n => patchSeriesWeight ε n / D n
have hδ (n : ℕ) : 0 < δ n := div_pos (patchSeriesWeight_pos hε n) (hD n)
refine ⟨δ, hδ, ?_⟩
intro n k hkn p
have hfn : ContDiffAt ℝ k (f n) p :=
(hf n).contDiffAt.of_le (WithTop.coe_le_coe.mpr le_top)
rw [iteratedFDeriv_const_smul_apply' hfn,
norm_smul, Real.norm_of_nonneg (hδ n).le]
calc
δ n * ‖iteratedFDeriv ℝ k (f n) p‖ ≤ δ n * D n :=
mul_le_mul_of_nonneg_left ((hC k n p).trans (hCD n k hkn)) (hδ n).le
_ = patchSeriesWeight ε n := div_mul_cancel₀ _ (hD n).ne'
omit [CompleteSpace V] in
theorem tsupport_const_smul_eq_of_pos (f : Coord → V) {δ : ℝ} (hδ : 0 < δ) :
tsupport (fun p => δ • f p) = tsupport f := by
change closure (Function.support (δ • f)) = closure (Function.support f)
rw [Function.support_const_smul_of_ne_zero δ f hδ.ne']
theorem exists_small_flat_accumulating_profile_sum (f : ℕ → Coord → V)
(hf : ∀ n, ContDiff ℝ ∞ (f n))
(hsupp : ∀ n, tsupport (f n) ⊆ enlargedAccumulatingDisk n)
(ε : ℝ) (hε : 0 < ε) :
∃ δ : ℕ → ℝ, (∀ n, 0 < δ n) ∧
(∀ n, tsupport (fun p => δ n • f n p) = tsupport (f n)) ∧
(∀ p, Summable (fun n => δ n • f n p)) ∧
ContDiff ℝ ∞ (flatPatchSum (fun n p => δ n • f n p)) ∧
(∀ p, ‖flatPatchSum (fun n q => δ n • f n q) p‖ ≤ ε) ∧
(∀ k : ℕ, iteratedFDeriv ℝ k
(flatPatchSum (fun n p => δ n • f n p)) (0 : Coord) = 0) := by
have hc := hasCompactSupport_of_accumulatingDisk_support f hsupp
obtain ⟨δ, hδ, hb⟩ := exists_positive_diagonal_amplitudes f hf hc ε hε
have hs (n : ℕ) : tsupport (fun p => δ n • f n p) = tsupport (f n) :=
tsupport_const_smul_eq_of_pos (f n) (hδ n)
have hsum := accumulatingDisk_flat_sum (fun n p => δ n • f n p) ε
(fun n => (hf n).const_smul (δ n))
(fun n => by rw [hs n]; exact hsupp n) hb
exact ⟨δ, hδ, hs, hsum.1, hsum.2.1, hsum.2.2.1, hsum.2.2.2⟩
end SmoothLocal.Geometry
end
end OAI
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