Theorems · Theorem · functional analysis

SchwartzMap.lineDerivOp_fourier_eq

∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {V : Type u_3} [inst_2 : NormedAddCommGroup V]
  [inst_3 : InnerProductSpace ℝ V] [inst_4 : FiniteDimensional ℝ V] [inst_5 : MeasurableSpace V] [inst_6 : BorelSpace V]
  (f : SchwartzMap V E) (m : V),
  LineDeriv.lineDerivOp m (FourierTransform.fourier f) =
    FourierTransform.fourier (-(2 * ↑Real.pi * Complex.I) • (SchwartzMap.smulLeftCLM E fun x => inner ℝ x m) f)

The line derivative in direction m of the Fourier transform is given by the Fourier transform of the multiplication with -(2 * π * Complex.I) • (inner ℝ · m).

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

NormedAddCommGroupNormedSpaceNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpace