openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Certificates /ContinuousSampledFrame.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzSampleContinuity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.SampledSchwartzFrame | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousLocalizationContinuity | |
| namespace OAI | |
| /-! # Continuous sampled frames and their finite coordinate matrices -/ | |
| open scoped SchwartzMap | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] | |
| attribute [local irreducible] sampledSchwartzFrame expandingCoordinates homogeneousLocalizationCLM | |
| theorem continuous_sampledSchwartzFrame (a k : ℝ) (ha1 : a < 1) (hk : 8 < k) | |
| (B : F →ₗ[ℝ] 𝓢(E, ℂ)) : Continuous (sampledSchwartzFrame a k ha1 hk B) := by | |
| classical | |
| apply continuous_clm_apply.mpr | |
| intro v | |
| have he (L : Radius) : sampledSchwartzFrame a k ha1 hk B L v = | |
| ∑ i, (Module.finBasis ℝ F).repr v i • | |
| schwartzTorusSample a k L.1 ha1 hk L.2 | |
| (radianFourierKernel (B ((Module.finBasis ℝ F) i))) := by | |
| conv_lhs => rw [← (Module.finBasis ℝ F).sum_repr v] | |
| simp only [map_sum, map_smul, sampledSchwartzFrame_basis] | |
| simp_rw [he] | |
| exact continuous_finsetSum _ (fun i _ => | |
| (continuous_const : Continuous (fun _ : Radius => (Module.finBasis ℝ F).repr v i)).smul | |
| (continuous_schwartzTorusSample a k ha1 hk _)) | |
| theorem continuous_sampledSchwartzFrame_coordinates (a k : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ)) | |
| (π : HomogeneousY a k →L[ℝ] F) (B : F →ₗ[ℝ] 𝓢(E, ℂ)) : | |
| Continuous (fun L : Radius => (expandingCoordinates a k ha ha1 hk χ π L).comp | |
| (sampledSchwartzFrame a k ha1 hk B L)) := by | |
| apply continuous_clm_apply.mpr | |
| intro v | |
| have hv : Continuous (fun L : Radius => sampledSchwartzFrame a k ha1 hk B L v) := | |
| (continuous_sampledSchwartzFrame a k ha1 hk B).clm_apply continuous_const | |
| exact (continuous_expandingCoordinates_apply a k ha ha1 hk χ π).comp | |
| (continuous_id.prodMk hv) | |
| end DefocusingNLS | |
| end OAI | |