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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 | import OAI.MeasureTheory.Falconer.Shells.HighShellCoefficientBounds
import OAI.MeasureTheory.Falconer.Angular.InverseCircleArcLength
import OAI.MeasureTheory.Falconer.Fourier.GraphKernelBounds
namespace OAI
namespace PlanarFalconer
open MeasureTheory
open scoped ComplexConjugate
namespace PositiveDistanceData
noncomputable def inverseDescriptorList (d : PositiveDistanceData) : Finset PolicyDescriptor :=
policyDescriptorList (fun n => d.mass n-(n:ℝ)/(d.N:ℝ)) d.N d.q d.a d.t
noncomputable def inverseEndpointSymbol (d : PositiveDistanceData) (k : PairMaskIndex → ℕ)
(l : Fin 2) : Plane → UnitCircle → ℂ :=
inheritedCircleSymbol d.inverseDescriptorList id (fun b => k (b,l)) d.Q d.ρ d.mass
d.N d.lam d.sig d.J d.i (fun _ => 1)
theorem inverseEndpointSymbol_measurable (d : PositiveDistanceData) (k : PairMaskIndex → ℕ)
(l : Fin 2) : Measurable (fun z : Plane × UnitCircle => d.inverseEndpointSymbol k l z.1 z.2) :=
measurable_inheritedCircleSymbol _ _ _ _ _ _ _ _ _ _ _ _ measurable_const
theorem inverseEndpointSymbol_norm (d : PositiveDistanceData) (hQ : 0 < d.Q.side) (j : ℕ)
{b : ℂ × (PairMaskIndex → ℕ)} (hb : b ∈ inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList))
(l : Fin 2) (x : Plane) (dir : UnitCircle) :
‖d.inverseEndpointSymbol b.2 l x dir‖ ≤ symbolDerivativeConstant (2*j)^(2*j) := by
exact inheritedCircleSymbol_norm_le _ _ _ _ hQ _ _ _ _ _ _ _ _ (2*j)
(pair_assignment_endpoint_budget _ _ _ (inverseLeibnizTerms_budget j _ hb) l) x (by simp) dir
theorem inverseCircleOperator_pair_terms (d : PositiveDistanceData) (hQ : 0 < d.Q.side)
(z : Plane × Plane) (j : ℕ) (dir : UnitCircle) :
inverseCircleOperator j (d.pairMaskSymbol z) dir =
((inverseLeibnizTerms j (pairMaskIndices d.inverseDescriptorList)).map (fun b =>
b.1 * ((assignmentScale d.inverseDescriptorList d.N d.sig b.2:ℂ) *
(d.inverseEndpointSymbol b.2 0 z.1 dir * conj (d.inverseEndpointSymbol b.2 1 z.2 dir))))).sum := by
have h := d.pairMaskSymbol_inverse_leibniz hQ z j (sourceAngle dir)
simpa only [inverseCircleOperator, sourceUnitDirection_sourceAngle,
inverseDescriptorList, inverseEndpointSymbol] using h
end PositiveDistanceData
end PlanarFalconer
end OAI
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