Theorems · Theorem · harmonic analysis

VectorFourier.fourierIntegral_iteratedFDeriv

∀ {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] {N : ℕ∞},
  ContDiff ℝ (↑N) f →
    (∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (iteratedFDeriv ℝ n f) μ) →
      ∀ {n : ℕ},
        ↑n ≤ N →
          VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L)
              (iteratedFDeriv ℝ n f) =
            fun w =>
            VectorFourier.fourierPowSMulRight (-L.flip)
              (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L) f) w n

The Fourier integral of the n-th derivative of a function is obtained by multiplying the Fourier integral of the original function by (2πI L w ⬝ )^n.

Defined in

Mathlib.Analysis.Fourier.FourierTransformDeriv

Cited by

2 results in Mathlib

Foundations

Depth 290 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure