| import OAI.Geometry.IsometricImmersion.Assembly.FlatPatchSum |
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| namespace OAI |
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| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Filter Set |
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| namespace SmoothLocal.Geometry |
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| variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V] |
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| 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' |
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| 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'] |
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| 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⟩ |
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| end SmoothLocal.Geometry |
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| end |
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| end OAI |
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