Theorems · Theorem · harmonic analysis

VectorFourier.contDiff_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] {μ : MeasureTheory.Measure V}
  [SecondCountableTopology V] {N : ℕ∞},
  (∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ) →
    ContDiff ℝ (↑N) (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L) f)

If ‖v‖^n * ‖f v‖ is integrable for all n ≤ N, then the Fourier transform of f is C^N.

Defined in

Mathlib.Analysis.Fourier.FourierTransformDeriv

Cited by

4 results in Mathlib

Foundations

Depth 267 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceSecondCountableTopology