Theorems · Theorem · harmonic analysis

VectorFourier.fourierIntegral_convergent_iff

∀ {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}
  [inst_6 : NormedAddCommGroup E] [inst_7 : NormedSpace ℂ E] [inst_8 : TopologicalSpace 𝕜] [IsTopologicalRing 𝕜]
  [inst_10 : TopologicalSpace V] [BorelSpace V] [inst_12 : TopologicalSpace W] {e : AddChar 𝕜 Circle}
  {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜},
  Continuous ⇑e →
    (Continuous fun p => (L p.1) p.2) →
      ∀ {f : V → E} (w : W), MeasureTheory.Integrable (fun v => e (-(L v) w) • f v) μ ↔ MeasureTheory.Integrable f μ

For any w, the Fourier integral is convergent iff f is integrable.

Defined in

Mathlib.Analysis.Fourier.FourierTransform

Cited by

9 results in Mathlib

Foundations

Depth 185 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleMeasurableSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceTopologicalSpaceIsTopologicalRingTopologicalSpaceBorelSpaceTopologicalSpace