Theorems · Theorem · functional analysis

TemperedDistribution.lineDerivOp_fourier_eq

∀ {E : Type u_3} {F : Type u_4} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : FiniteDimensional ℝ E] [inst_3 : MeasurableSpace E] [inst_4 : BorelSpace E] [inst_5 : NormedAddCommGroup F]
  [inst_6 : NormedSpace ℂ F] (f : TemperedDistribution E F) (m : E),
  LineDeriv.lineDerivOp m (FourierTransform.fourier f) =
    FourierTransform.fourier
      (-(2 * ↑Real.pi * Complex.I) • (TemperedDistribution.smulLeftCLM F 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.TemperedDistribution

Cited by

0 results in Mathlib

Foundations

Depth 302 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceNormedAddCommGroupNormedSpace