Theorems · Theorem · harmonic analysis

VectorFourier.hasFDerivAt_fourierIntegral

∀ {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] [SecondCountableTopology V]
  {μ : MeasureTheory.Measure V},
  MeasureTheory.Integrable f μ →
    MeasureTheory.Integrable (fun v => ‖v‖ * ‖f v‖) μ →
      ∀ (w : W),
        HasFDerivAt (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
          (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L)
            (VectorFourier.fourierSMulRight L f) w)
          w

Main theorem of this section: if both f and x ↦ ‖x‖ * ‖f x‖ are integrable, then the Fourier transform of f has a Fréchet derivative (everywhere in its domain) and its derivative is the Fourier transform of smulRight L f.

Defined in

Mathlib.Analysis.Fourier.FourierTransformDeriv

Cited by

5 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceSecondCountableTopology