File size: 3,244 Bytes
7f37411 | 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 59 60 61 62 63 64 65 66 67 68 69 | import OAI.MeasureTheory.Falconer.Shells.FiniteSpatialShell
namespace OAI
/-! # Square integrability of the finite stationary sum
The radial coefficients need only be bounded almost everywhere on the
source shell. No global bound at zero or negative frequency is required.
-/
namespace Falconer
open Set MeasureTheory
theorem finite_weighted_sum_sq_integrable {ι : Type*} (S : Finset ι)
(μ : Measure ℝ) [IsFiniteMeasure μ] (c F : ι → ℝ → ℂ)
(hc : ∀ i ∈ S, Measurable (c i)) (hF : ∀ i ∈ S, Measurable (F i))
(A B : ι → ℝ) (hA : ∀ i ∈ S, 0 ≤ A i) (hB : ∀ i ∈ S, 0 ≤ B i)
(hbF : ∀ i ∈ S, ∀ r, ‖F i r‖ ≤ B i)
(hbC : ∀ᵐ r ∂μ, ∀ i ∈ S, ‖c i r‖ ≤ A i) :
Integrable (fun r => ‖∑ i ∈ S, c i r*F i r‖^2) μ := by
have hm : Measurable (fun r => ∑ i ∈ S, c i r*F i r) :=
Finset.measurable_sum S (fun i hi => (hc i hi).mul (hF i hi))
let K := ∑ i ∈ S, A i*B i
have hK : 0 ≤ K := Finset.sum_nonneg (fun i hi => mul_nonneg (hA i hi) (hB i hi))
apply Integrable.of_bound (hm.norm.pow_const 2).aestronglyMeasurable (K^2)
filter_upwards [hbC] with r hr
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
apply (sq_le_sq₀ (norm_nonneg _) hK).mpr
apply (norm_sum_le S (fun i => c i r*F i r)).trans
apply Finset.sum_le_sum
intro i hi
rw [norm_mul]
exact mul_le_mul (hr i hi) (hbF i hi r) (norm_nonneg _) (hA i hi)
theorem finite_spatial_shell_sq_integrable {d : ℕ} {ι : Type*} (S : Finset ι)
(v : ℝ) (Ψ : ℝ → ℝ)
[IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)]
(δ L : ℝ) (hδ : 0 < δ) (hL : 0 < L) (j : ι → ℕ)
(U V : ι → Space d × (ℝ × Sphere d) → ℂ)
(hU : ∀ i ∈ S, Measurable (U i)) (hV : ∀ i ∈ S, Measurable (V i))
(B C : ι → ℝ) (hB : ∀ i ∈ S, 0 ≤ B i) (hC : ∀ i ∈ S, 0 ≤ C i)
(hbU : ∀ i ∈ S, ∀ z, ‖U i z‖ ≤ B i)
(hbV : ∀ i ∈ S, ∀ z, ‖V i z‖ ≤ C i)
(c : ι → ℝ → ℂ) (A : ι → ℝ)
(hc : ∀ i ∈ S, Measurable (c i)) (hA : ∀ i ∈ S, 0 ≤ A i)
(hbound : ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ,
∀ i ∈ S, ‖c i r‖ ≤ A i) :
Integrable (fun r => ‖∑ i ∈ S, c i r *
spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i) r‖^2)
(PlanarFalconer.sourceAngularShellMeasure v Ψ) := by
refine finite_weighted_sum_sq_integrable S _ c
(fun i => spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i)) hc ?_
A (fun i => ((∫ ξ : Space d, ‖spatialFourierCoeff δ L hδ hL (j i) ξ‖)*
(B i*C i))*(sphereMeasure d).real univ) hA ?_ ?_ hbound
· intro i hi
exact spatialSharedAmplitude_measurable _
(spatialFourierCoeff (d := d) δ L hδ hL (j i)).continuous.measurable
(U i) (V i) (hU i hi) (hV i hi)
· intro i hi
exact mul_nonneg (mul_nonneg (integral_nonneg (fun _ => norm_nonneg _))
(mul_nonneg (hB i hi) (hC i hi))) measureReal_nonneg
· intro i hi r
exact spatialSharedAmplitude_norm_le _ (spatialFourierCoeff_integrable δ L hδ hL (j i))
(U i) (V i) (hB i hi) (hbU i hi) (hbV i hi) r
end Falconer
end OAI
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