File size: 2,128 Bytes
d5b562f | 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 | 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
|