File size: 4,124 Bytes
d4317d7 | 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 81 82 83 84 | 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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