Theorems · Theorem · harmonic analysis

VectorFourier.fourierIntegral_fderiv

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {V : Type u_2} {W : Type u_3}
  [inst_2 : NormedAddCommGroup V] [inst_3 : NormedSpace ℝ V] [inst_4 : NormedAddCommGroup W] [inst_5 : NormedSpace ℝ W]
  (L : V →L[ℝ] W →L[ℝ] ℝ) {f : V → E} [inst_6 : MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V]
  {μ : MeasureTheory.Measure V} [μ.IsAddHaarMeasure],
  MeasureTheory.Integrable f μ →
    Differentiable ℝ f →
      MeasureTheory.Integrable (fderiv ℝ f) μ →
        VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L) (fderiv ℝ f) =
          VectorFourier.fourierSMulRight (-L.flip)
            (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L) f)

The Fourier integral of the derivative of a function is obtained by multiplying the Fourier integral of the original function by -L w v.

Defined in

Mathlib.Analysis.Fourier.FourierTransformDeriv

Cited by

2 results in Mathlib

Foundations

Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure