Theorems · Theorem · harmonic analysis

VectorFourier.integral_fourierIntegral_smul_eq_flip

∀ {𝕜 : Type u_1} [inst : CommRing 𝕜] {V : Type u_2} [inst_1 : AddCommGroup V] [inst_2 : Module 𝕜 V]
  [inst_3 : MeasurableSpace V] {W : Type u_3} [inst_4 : AddCommGroup W] [inst_5 : Module 𝕜 W] {F : Type u_5}
  [inst_6 : NormedAddCommGroup F] [inst_7 : NormedSpace ℂ F] [inst_8 : TopologicalSpace 𝕜] [IsTopologicalRing 𝕜]
  [inst_10 : TopologicalSpace V] [BorelSpace V] [inst_12 : TopologicalSpace W] [inst_13 : MeasurableSpace W]
  [BorelSpace W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜}
  {ν : MeasureTheory.Measure W} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
  [SecondCountableTopologyEither W V] [CompleteSpace F] {f : V → ℂ} {g : W → F},
  Continuous ⇑e →
    (Continuous fun p => (L p.1) p.2) →
      MeasureTheory.Integrable f μ →
        MeasureTheory.Integrable g ν →
          ∫ (ξ : W), VectorFourier.fourierIntegral e μ L f ξ • g ξ ∂ν =
            ∫ (x : V), f x • VectorFourier.fourierIntegral e ν L.flip g x ∂μ

The Fourier transform satisfies ∫ 𝓕 f * g = ∫ f * 𝓕 g, i.e., it is self-adjoint.

Defined in

Mathlib.Analysis.Fourier.FourierTransform

Cited by

1 results in Mathlib

Foundations

Depth 269 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleMeasurableSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceIsTopologicalRingTopologicalSpaceBorelSpaceTopologicalSpaceMeasurableSpaceBorelSpaceMeasureTheory.SigmaFiniteMeasureTheory.SigmaFiniteSecondCountableTopologyEitherCompleteSpace