| import OAI.MeasureTheory.Falconer.Graphs.GraphAmplitudeDerivatives |
| import OAI.MeasureTheory.Falconer.Fourier.GraphActualFrequencySupport |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set Metric Filter MeasureTheory |
| open scoped ContDiff Topology |
|
|
| def graphAmplitudeCompact : Set (Plane × ℝ) := |
| (closedBall 0 8 ×ˢ Icc (-4) 4) ∩ {p | 1/8 ≤ ‖p.1‖} |
|
|
| theorem graphAmplitudeCompact_isCompact : IsCompact graphAmplitudeCompact := |
| ((isCompact_closedBall (0:Plane) 8).prod isCompact_Icc).inter_right |
| (isClosed_le continuous_const continuous_fst.norm) |
|
|
| theorem graphAmplitudePullback_norm_transport (b : Fin 4 → Plane → UnitCircle → ℂ) |
| (x : Fin 4 → Plane) (distance plus : Bool) (axis δ : ℝ) (j k : ℕ) |
| (Ψ : ℝ → ℝ) (R : ℝ) (p : Plane × ℝ) : |
| ‖graphAmplitudePullback b x distance plus axis δ j k Ψ p‖ = |
| ‖graphTransport (graphFourKernel b (graphSourceWeight distance plus axis (angularBinCount δ) j k) x) |
| Ψ R (graphSourceChartAxis distance plus) (p.2,p.1)‖ := by |
| simp only [graphAmplitudePullback,graphShellJacobian,graphTransport,graphFourKernel, |
| norm_smul,norm_mul,sourcePlaneWave_norm,one_mul,mul_one,Complex.norm_real] |
| ring |
|
|
| theorem graphAmplitudePullback_tsupport (b : Fin 4 → Plane → UnitCircle → ℂ) |
| (x : Fin 4 → Plane) (distance plus : Bool) (axis δ : ℝ) (j k : ℕ) |
| {Ψ : ℝ → ℝ} (hsupp : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) : |
| tsupport (graphAmplitudePullback b x distance plus axis δ j k Ψ) ⊆ graphAmplitudeCompact := by |
| apply closure_minimal _ graphAmplitudeCompact_isCompact.isClosed |
| intro p hp |
| have hn : graphTransport |
| (graphFourKernel b (graphSourceWeight distance plus axis (angularBinCount δ) j k) x) |
| Ψ 1 (graphSourceChartAxis distance plus) (p.2,p.1) ≠ 0 := by |
| intro he |
| apply hp |
| apply norm_eq_zero.mp |
| rw [graphAmplitudePullback_norm_transport b x distance plus axis δ j k Ψ 1 p,he,norm_zero] |
| have hb := graphActualTransport_frequency_support b x distance plus axis (angularBinCount δ) j k hsupp 1 (p.2,p.1) hn |
| refine ⟨⟨?_,abs_le.mp hb.2.2⟩,hb.1⟩ |
| simpa only [mem_closedBall,dist_zero_right] using hb.2.1 |
|
|
| theorem graphAmplitudePullback_hasCompactSupport (b : Fin 4 → Plane → UnitCircle → ℂ) |
| (x : Fin 4 → Plane) (distance plus : Bool) (axis δ : ℝ) (j k : ℕ) |
| {Ψ : ℝ → ℝ} (hsupp : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) : |
| HasCompactSupport (graphAmplitudePullback b x distance plus axis δ j k Ψ) := |
| graphAmplitudeCompact_isCompact.of_isClosed_subset isClosed_closure |
| (graphAmplitudePullback_tsupport b x distance plus axis δ j k hsupp) |
|
|
| theorem GraphDerivativeBound.global_smooth {n : ℕ} {f : (Plane × ℝ) → ℂ} {C D : ℝ} |
| (hf : GraphDerivativeBound n f C D) (hs : tsupport f ⊆ graphAmplitudeCompact) : |
| ContDiff ℝ ∞ f := by |
| rw [contDiff_iff_contDiffAt] |
| intro p |
| by_cases hp : p.1 ≠ 0 |
| · exact (hf.smooth p hp).contDiffAt (graphSmoothDomain_isOpen.mem_nhds hp) |
| · have hn : p ∉ tsupport f := by |
| intro hmem |
| have hlo : (1:ℝ)/8 ≤ ‖p.1‖ := (hs hmem).2 |
| rw [not_ne_iff.mp hp,norm_zero] at hlo |
| norm_num at hlo |
| apply (contDiffAt_const (c := (0:ℂ))).congr_of_eventuallyEq |
| filter_upwards [isClosed_closure.isOpen_compl.mem_nhds hn] with q hq |
| exact image_eq_zero_of_notMem_tsupport hq |
|
|
| theorem GraphDerivativeBound.global_bound {n : ℕ} {f : (Plane × ℝ) → ℂ} {C D : ℝ} |
| (hf : GraphDerivativeBound n f C D) (hs : tsupport f ⊆ graphAmplitudeCompact) |
| (hC : 0 ≤ C) (hD : 0 ≤ D) (j : ℕ) (hj : j ≤ n) (p : Plane × ℝ) : |
| ‖iteratedFDeriv ℝ j f p‖ ≤ C*D^j := by |
| by_cases hp : p ∈ graphAmplitudeCompact |
| · apply hf.bound j hj p.1 p.2 _ (abs_le.mpr hp.1.2) hp.2 |
| simpa only [mem_closedBall,dist_zero_right] using hp.1.1 |
| · have hz : iteratedFDeriv ℝ j f p = 0 := by |
| apply Function.notMem_support.mp |
| intro hmem |
| exact hp (hs (support_iteratedFDeriv_subset j hmem)) |
| rw [hz,norm_zero] |
| exact mul_nonneg hC (pow_nonneg hD j) |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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