| import OAI.Geometry.IsometricImmersion.Pulses.ActualForcingMoment |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.Weighted |
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|
| theorem pulseTest_abs_le (a tau x : ℝ) : |pulseTest a tau x| ≤ axisBump a x := by |
| unfold pulseTest |
| rw [abs_mul,abs_of_nonneg (axisBump_nonneg a x)] |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left (Real.abs_cos_le_one _) (axisBump_nonneg a x) |
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|
| theorem axisBump_Icc_integral {a : ℝ} (ha : 0 < a) : |
| (∫ x in Icc (-a) a, axisBump a x) = ∫ x : ℝ, axisBump a x := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro x hx |
| by_contra hn |
| have h := abs_lt.mp (axisBump_support_bound ha hn) |
| exact hx ⟨h.1.le,h.2.le⟩ |
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|
| theorem pulse_spatial_mass_integral {a delta tau : ℝ} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) : |
| (∫ q in pulsePairRectangle a delta tau, axisBump a q.2 ∂volume.prod volume) = |
| (2*delta/tau)*(∫ x : ℝ, axisBump a x) := by |
| have hw : -(delta/tau) ≤ delta/tau := by linarith [div_nonneg hd ht.le] |
| have hh := setIntegral_prod_mul (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ)) |
| (fun _ : ℝ => (1 : ℝ)) (axisBump a) (Icc (-(delta/tau)) (delta/tau)) (Icc (-a) a) |
| simp only [one_mul] at hh |
| rw [show pulsePairRectangle a delta tau = |
| Icc (-(delta/tau)) (delta/tau) ×ˢ Icc (-a) a from rfl,hh, |
| setIntegral_const,Real.volume_real_Icc_of_le hw,smul_eq_mul,mul_one,axisBump_Icc_integral ha] |
| ring |
|
|
| theorem pulseWeightedMoment_bound {a delta tau B : ℝ} {f : Coord → ℝ} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) (hB : 0 ≤ B) |
| (hf : ∀ p ∈ pulseStrip a delta tau, |f p| ≤ B) : |
| |pulseWeightedMoment a delta tau f| ≤ |
| B*(2*delta/tau)*(∫ x : ℝ, axisBump a x) := by |
| have hmajor : IntegrableOn (fun q : ℝ × ℝ => B*axisBump a q.2) |
| (pulsePairRectangle a delta tau) (volume.prod volume) := |
| ((continuous_const.mul ((axisBump_contDiff a).continuous.comp continuous_snd)).continuousOn).integrableOn_compact |
| (pulsePairRectangle_isCompact a delta tau) |
| have hpoint : ∀ᵐ q ∂(volume.prod volume).restrict (pulsePairRectangle a delta tau), |
| ‖f (pulsePairPoint q)*pulseTest a tau q.2‖ ≤ B*axisBump a q.2 := by |
| filter_upwards [ae_restrict_mem (pulsePairRectangle_isCompact a delta tau).measurableSet] with q hq |
| rw [Real.norm_eq_abs,abs_mul] |
| exact mul_le_mul (hf _ (pulsePairPoint_mem hq)) (pulseTest_abs_le a tau q.2) |
| (abs_nonneg _) hB |
| have h := norm_integral_le_of_norm_le hmajor hpoint |
| simp only [Real.norm_eq_abs,integral_const_mul] at h |
| rw [pulse_spatial_mass_integral ha hd ht] at h |
| simpa only [pulseWeightedMoment,mul_assoc] using h |
|
|
| theorem pulseWeightedMoment_integrable {a delta tau : ℝ} {f : Coord → ℝ} |
| (hf : ContinuousOn f (pulseStrip a delta tau)) : |
| IntegrableOn (fun q => f (pulsePairPoint q)*pulseTest a tau q.2) |
| (pulsePairRectangle a delta tau) (volume.prod volume) := |
| ((hf.comp pulsePairPoint_continuous.continuousOn (fun _ hq => pulsePairPoint_mem hq)).mul |
| ((pulseTest_continuous a tau).comp continuous_snd).continuousOn).integrableOn_compact |
| (pulsePairRectangle_isCompact a delta tau) |
|
|
| theorem pulseWeightedMoment_sub {a delta tau : ℝ} {f h : Coord → ℝ} |
| (hf : ContinuousOn f (pulseStrip a delta tau)) (hh : ContinuousOn h (pulseStrip a delta tau)) : |
| pulseWeightedMoment a delta tau (fun p => f p-h p) = |
| pulseWeightedMoment a delta tau f-pulseWeightedMoment a delta tau h := by |
| unfold pulseWeightedMoment |
| simp_rw [sub_mul] |
| exact integral_sub (pulseWeightedMoment_integrable hf) (pulseWeightedMoment_integrable hh) |
|
|
| theorem pulseWeightedMoment_lower_of_remainder {a delta tau B L : ℝ} {f h : Coord → ℝ} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) (hB : 0 ≤ B) |
| (hf : ContinuousOn f (pulseStrip a delta tau)) (hh : ContinuousOn h (pulseStrip a delta tau)) |
| (hrem : ∀ p ∈ pulseStrip a delta tau, |f p-h p| ≤ B) |
| (hlead : L ≤ |pulseWeightedMoment a delta tau h|) : |
| L-B*(2*delta/tau)*(∫ x : ℝ, axisBump a x) ≤ |pulseWeightedMoment a delta tau f| := by |
| have herr := pulseWeightedMoment_bound ha hd ht hB hrem |
| rw [pulseWeightedMoment_sub hf hh] at herr |
| have htriangle : |pulseWeightedMoment a delta tau h| ≤ |
| |pulseWeightedMoment a delta tau f|+ |
| |pulseWeightedMoment a delta tau f-pulseWeightedMoment a delta tau h| := by |
| calc |
| _ = |pulseWeightedMoment a delta tau f- |
| (pulseWeightedMoment a delta tau f-pulseWeightedMoment a delta tau h)| := by congr 1; ring |
| _ ≤ _ := abs_sub _ _ |
| linarith |
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| theorem pulseWeightedMoment_lower_order_error {a delta tau C : ℝ} {N : ℕ} {f : Coord → ℝ} |
| (ha : 0 < a) (hd : 0 ≤ delta) (ht : 0 < tau) (hC : 0 ≤ C) |
| (hf : ∀ p ∈ pulseStrip a delta tau, |f p| ≤ C*tau/tau^N) : |
| |pulseWeightedMoment a delta tau f| ≤ |
| (2*C*(∫ x : ℝ, axisBump a x))*delta/tau^N := by |
| have hB : 0 ≤ C*tau/tau^N := div_nonneg (mul_nonneg hC ht.le) (pow_nonneg ht.le N) |
| apply (pulseWeightedMoment_bound ha hd ht hB hf).trans_eq |
| field_simp [ht.ne'] |
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| end SmoothLocal.Pulse |
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| end |
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| end OAI |
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