| import Mathlib |
| import OAI.Analysis.BiholderTransport.Calculus.TaylorBound |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Asymptotics |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
| variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
|
|
| def HasSecondTaylor (f : E → ℝ) (l : E →L[ℝ] ℝ) |
| (B : E →L[ℝ] E →L[ℝ] ℝ) : Prop := |
| (fun h => f h-f 0-l h-B h h/2) =o[𝓝 0] (fun h : E => ‖h‖^2) |
|
|
| lemma HasSecondTaylor.eventually {f : E → ℝ} {l : E →L[ℝ] ℝ} |
| {B : E →L[ℝ] E →L[ℝ] ℝ} (hf : HasSecondTaylor f l B) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ h in 𝓝 (0:E), |f h-f 0-l h-B h h/2|≤ε*‖h‖^2 := by |
| simpa only [Real.norm_eq_abs,abs_pow,abs_norm] using hf.def hε |
|
|
| lemma hasSecondTaylor_of_contDiffAt {f : E → ℝ} (hf : ContDiffAt ℝ 2 f 0) : |
| HasSecondTaylor f (fderiv ℝ f 0) (fderiv ℝ (fderiv ℝ f) 0) := by |
| apply Asymptotics.IsLittleO.of_bound |
| intro ε hε |
| have hn := hf.eventually (by norm_num) |
| have hD := ((hf.fderiv_right (m := 1) (by norm_num)).fderiv_right |
| (m := 0) (by norm_num)).continuousAt |
| have he : ∀ᶠ z : E in 𝓝 0, |
| ‖fderiv ℝ (fderiv ℝ f) z-fderiv ℝ (fderiv ℝ f) 0‖<ε := by |
| exact Metric.tendsto_nhds.mp hD ε hε |>.mono (fun _ h => by simpa only [dist_eq_norm] using h) |
| obtain ⟨r,hr,hball⟩ := Metric.eventually_nhds_iff.mp (hn.and he) |
| filter_upwards [Metric.ball_mem_nhds (0:E) hr] with h hh |
| have H := second_order_remainder_bound_on_ball hr hε.le |
| (fun z hz => (hball hz).1) (fun z hz => (hball hz).2.le) hh |
| simpa only [Real.norm_eq_abs,abs_pow,abs_norm] using H |
|
|
| lemma bilinear_isBigO_norm_mul {α : Type*} (L : Filter α) |
| (B : E →L[ℝ] F →L[ℝ] G) (u : α → E) (v : α → F) : |
| (fun x => B (u x) (v x)) =O[L] (fun x => ‖u x‖*‖v x‖) := by |
| apply Asymptotics.IsBigO.of_bound ‖B‖ |
| exact Filter.Eventually.of_forall (fun x => by |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (norm_nonneg _) (norm_nonneg _)), |
| mul_assoc] using B.le_opNorm₂ (u x) (v x)) |
|
|
| lemma bilinear_remainder_of_hasFDerivAt {a : E → F} {A : E →L[ℝ] F} |
| (ha : HasFDerivAt a A 0) (ha0 : a 0=0) (B : F →L[ℝ] F →L[ℝ] ℝ) : |
| (fun h => B (a h) (a h)-B (A h) (A h)) =o[𝓝 0] (fun h : E => ‖h‖^2) := by |
| have hR : (fun h => a h-A h) =o[𝓝 0] (fun h : E => h) := by |
| simpa only [zero_add,ha0,sub_zero] using hasFDerivAt_iff_isLittleO_nhds_zero.mp ha |
| have hO : a =O[𝓝 0] (fun h : E => h) := by |
| simpa only [ha0,sub_zero] using ha.isBigO_sub |
| have hA : (fun h : E => A h) =O[𝓝 0] (fun h => h) := A.isBigO_id _ |
| have h1 := (bilinear_isBigO_norm_mul (𝓝 (0:E)) B (fun h => a h-A h) a).trans_isLittleO |
| (hR.norm_left.norm_right.mul_isBigO hO.norm_left.norm_right) |
| have h2 := (bilinear_isBigO_norm_mul (𝓝 (0:E)) B (fun h => A h) (fun h => a h-A h)).trans_isLittleO |
| (hA.norm_left.norm_right.mul_isLittleO hR.norm_left.norm_right) |
| have he : (fun h => B (a h) (a h)-B (A h) (A h)) = |
| (fun h => B (a h-A h) (a h)+B (A h) (a h-A h)) := by |
| funext h |
| simp only [map_sub,_root_.sub_apply] |
| ring |
| rw [he] |
| simpa only [pow_two] using h1.add h2 |
|
|
| lemma HasSecondTaylor.comp_stationary {f : F → ℝ} |
| {B : F →L[ℝ] F →L[ℝ] ℝ} (hf : HasSecondTaylor f 0 B) |
| {a : E → F} {A : E →L[ℝ] F} (ha : HasFDerivAt a A 0) (ha0 : a 0=0) : |
| HasSecondTaylor (fun h => f (a h)) 0 |
| (((ContinuousLinearMap.compL ℝ E F ℝ).flip A).comp (B.comp A)) := by |
| have hO : (fun h : E => ‖a h‖) =O[𝓝 0] (fun h => ‖h‖) := by |
| simpa only [ha0,sub_zero] using ha.isBigO_sub.norm_left.norm_right |
| have ht : Tendsto a (𝓝 0) (𝓝 0) := ha0 ▸ ha.continuousAt |
| have h1 := (hf.comp_tendsto ht).trans_isBigO (hO.pow 2) |
| have h2 : (fun h => (B (a h) (a h)-B (A h) (A h))/2) =o[𝓝 0] |
| (fun h : E => ‖h‖^2) := by |
| simpa only [div_eq_mul_inv,mul_comm] using |
| (bilinear_remainder_of_hasFDerivAt ha ha0 B).const_mul_left (2:ℝ)⁻¹ |
| unfold HasSecondTaylor |
| have he : (fun h => f (a h)-f (a 0)-(0 : E →L[ℝ] ℝ) h- |
| (((ContinuousLinearMap.compL ℝ E F ℝ).flip A).comp (B.comp A)) h h/2) = |
| (fun h => (f (a h)-f 0-(0 : F →L[ℝ] ℝ) (a h)-B (a h) (a h)/2)+ |
| (B (a h) (a h)-B (A h) (A h))/2) := by |
| funext h |
| simp only [ha0,_root_.zero_apply,sub_zero,ContinuousLinearMap.comp_apply, |
| ContinuousLinearMap.compL_apply,ContinuousLinearMap.flip_apply] |
| ring |
| rw [he] |
| exact h1.add h2 |
|
|
| lemma HasSecondTaylor.sub {f g : E → ℝ} {l k : E →L[ℝ] ℝ} |
| {B C : E →L[ℝ] E →L[ℝ] ℝ} (hf : HasSecondTaylor f l B) |
| (hg : HasSecondTaylor g k C) : HasSecondTaylor (fun h => f h-g h) (l-k) (B-C) := by |
| unfold HasSecondTaylor at * |
| have he : (fun h => (f h-g h)-(f 0-g 0)-(l-k) h-(B-C) h h/2) = |
| (fun h => (f h-f 0-l h-B h h/2)-(g h-g 0-k h-C h h/2)) := by |
| funext h |
| simp only [_root_.sub_apply] |
| ring |
| rw [he] |
| exact Asymptotics.IsLittleO.sub hf hg |
|
|
| lemma HasSecondTaylor.add {f g : E → ℝ} {l k : E →L[ℝ] ℝ} |
| {B C : E →L[ℝ] E →L[ℝ] ℝ} (hf : HasSecondTaylor f l B) |
| (hg : HasSecondTaylor g k C) : HasSecondTaylor (fun h => f h+g h) (l+k) (B+C) := by |
| unfold HasSecondTaylor at * |
| have he : (fun h => (f h+g h)-(f 0+g 0)-(l+k) h-(B+C) h h/2) = |
| (fun h => (f h-f 0-l h-B h h/2)+(g h-g 0-k h-C h h/2)) := by |
| funext h |
| simp only [_root_.add_apply] |
| ring |
| rw [he] |
| exact Asymptotics.IsLittleO.add hf hg |
|
|
| end WeakMTWTransport |
|
|
| end |
|
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| end OAI |
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