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ef96466 | 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 | 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
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