Theorems · Theorem · functional analysis

SchwartzMap.integral_fourier_smul_eq

∀ {V : Type u_3} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : FiniteDimensional ℝ V]
  [inst_3 : MeasurableSpace V] [inst_4 : BorelSpace V] {F : Type u_4} [inst_5 : NormedAddCommGroup F]
  [inst_6 : NormedSpace ℂ F] [CompleteSpace F] (f : SchwartzMap V ℂ) (g : SchwartzMap V F),
  ∫ (ξ : V), (FourierTransform.fourier f) ξ • g ξ = ∫ (x : V), f x • (FourierTransform.fourier g) x

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

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Cited by

2 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceNormedAddCommGroupNormedSpaceCompleteSpace