Theorems · Theorem · functional analysis

SchwartzMap.fderivCLM_fourier_eq

∀ (𝕜 : Type u_1) [inst : RCLike 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℂ E]
  [inst_3 : NormedSpace 𝕜 E] [inst_4 : SMulCommClass ℂ 𝕜 E] {V : Type u_3} [inst_5 : NormedAddCommGroup V]
  [inst_6 : InnerProductSpace ℝ V] [inst_7 : FiniteDimensional ℝ V] [inst_8 : MeasurableSpace V] [inst_9 : BorelSpace V]
  (f : SchwartzMap V E),
  (SchwartzMap.fderivCLM 𝕜 V E) (FourierTransform.fourier f) =
    FourierTransform.fourier (-(2 * ↑Real.pi * Complex.I) • (SchwartzMap.smulRightCLM ℂ E (innerSL ℝ)) f)

The derivative of the Fourier transform is given by the Fourier transform of the multiplication with -(2 * π * Complex.I) • innerSL ℝ.

Defined in

Mathlib.Analysis.Distribution.SchwartzSpace.Fourier

Cited by

1 results in Mathlib

Foundations

Depth 298 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedSpaceSMulCommClassNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpace