Theorems · Theorem · harmonic analysis

VectorFourier.integral_sesq_fourierIntegral_eq_neg_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] {E : Type u_4}
  {F : Type u_5} {G : Type u_6} [inst_6 : NormedAddCommGroup E] [inst_7 : NormedSpace ℂ E]
  [inst_8 : NormedAddCommGroup F] [inst_9 : NormedSpace ℂ F] [inst_10 : NormedAddCommGroup G]
  [inst_11 : NormedSpace ℂ G] [inst_12 : TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [inst_14 : TopologicalSpace V]
  [BorelSpace V] [inst_16 : TopologicalSpace W] [inst_17 : 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 E] [CompleteSpace F] {f : V → E}
  {g : W → F} (M : E →L⋆[ℂ] F →L[ℂ] G),
  Continuous ⇑e →
    (Continuous fun p => (L p.1) p.2) →
      MeasureTheory.Integrable f μ →
        MeasureTheory.Integrable g ν →
          ∫ (ξ : W), (M (VectorFourier.fourierIntegral e μ L f ξ)) (g ξ) ∂ν =
            ∫ (x : V), (M (f x)) (VectorFourier.fourierIntegral e ν (-L.flip) g x) ∂μ

The Fourier transform satisfies ∫ 𝓕 f * conj g = ∫ f * conj (𝓕⁻¹ g), which together with the Fourier inversion theorem yields Plancherel's theorem. The stated version is more convenient since it does only require integrability of f and g. Version where the multiplication is replaced by a general bilinear form M.

Defined in

Mathlib.Analysis.Fourier.FourierTransform

Cited by

1 results in Mathlib

Foundations

Depth 268 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleMeasurableSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceIsTopologicalRingTopologicalSpaceBorelSpaceTopologicalSpaceMeasurableSpaceBorelSpaceMeasureTheory.SigmaFiniteMeasureTheory.SigmaFiniteSecondCountableTopologyEitherCompleteSpaceCompleteSpace