openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralBoundedCoefficientAction.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralFieldParameterLimit | |
| namespace OAI | |
| /-! Bounded coefficient errors acting on the approximate outgoing columns. | |
| Vanishing coefficient errors give vanishing normalized forcing. -/ | |
| open Filter Topology | |
| open scoped BoundedContinuousFunction | |
| namespace DefocusingNLS | |
| local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ) | |
| noncomputable def boundedCircularCoefficientAction (A B : ℝ →ᵇ ℂ) (Z : CircularTailSpace) : | |
| CircularTailSpace := by | |
| let N := fun t => circularCoefficientAction (A t) (B t) (circularTailEvaluation Z t) | |
| have hc : Continuous N := by | |
| dsimp [N,circularCoefficientAction,circularTailEvaluation] | |
| fun_prop | |
| have hb (t : ℝ) : ‖N t‖ ≤ (‖A‖+‖B‖)*‖Z‖ := by | |
| exact (circularCoefficientAction_norm _ _ _).trans | |
| (mul_le_mul (add_le_add (A.norm_coe_le_norm t) (B.norm_coe_le_norm t)) | |
| (circularTailEvaluation_norm Z t) (norm_nonneg _) (by positivity)) | |
| exact (BoundedContinuousFunction.ofNormedAddCommGroup (fun t => (N t).1) | |
| hc.fst ((‖A‖+‖B‖)*‖Z‖) (fun t => (norm_fst_le (N t)).trans (hb t)), | |
| BoundedContinuousFunction.ofNormedAddCommGroup (fun t => (N t).2) | |
| hc.snd ((‖A‖+‖B‖)*‖Z‖) (fun t => (norm_snd_le (N t)).trans (hb t))) | |
| theorem boundedCircularCoefficientAction_evaluation (A B : ℝ →ᵇ ℂ) (Z : CircularTailSpace) (t : ℝ) : | |
| circularTailEvaluation (boundedCircularCoefficientAction A B Z) t= | |
| circularCoefficientAction (A t) (B t) (circularTailEvaluation Z t) := rfl | |
| theorem boundedCircularCoefficientAction_norm (A B : ℝ →ᵇ ℂ) (Z : CircularTailSpace) : | |
| ‖boundedCircularCoefficientAction A B Z‖ ≤ (‖A‖+‖B‖)*‖Z‖ := by | |
| have hb (t : ℝ) : ‖circularCoefficientAction (A t) (B t) (circularTailEvaluation Z t)‖ ≤ | |
| (‖A‖+‖B‖)*‖Z‖ := by | |
| exact (circularCoefficientAction_norm _ _ _).trans | |
| (mul_le_mul (add_le_add (A.norm_coe_le_norm t) (B.norm_coe_le_norm t)) | |
| (circularTailEvaluation_norm Z t) (norm_nonneg _) (by positivity)) | |
| apply max_le | |
| · apply (BoundedContinuousFunction.norm_le (by positivity)).mpr | |
| intro t | |
| exact (norm_fst_le _).trans (hb t) | |
| · apply (BoundedContinuousFunction.norm_le (by positivity)).mpr | |
| intro t | |
| exact (norm_snd_le _).trans (hb t) | |
| theorem boundedCircularCoefficientAction_tendsto_zero (A B : ℕ → ℝ →ᵇ ℂ) | |
| (Z : ℕ → CircularTailSpace) (Z₀ : CircularTailSpace) | |
| (hA : Tendsto A atTop (𝓝 0)) (hB : Tendsto B atTop (𝓝 0)) | |
| (hZ : Tendsto Z atTop (𝓝 Z₀)) : | |
| Tendsto (fun n => boundedCircularCoefficientAction (A n) (B n) (Z n)) atTop (𝓝 0) := by | |
| apply squeeze_zero_norm (fun n => boundedCircularCoefficientAction_norm (A n) (B n) (Z n)) | |
| simpa only [norm_zero,add_zero,zero_mul] using (hA.norm.add hB.norm).mul hZ.norm | |
| end DefocusingNLS | |
| end OAI | |