Theorems · Theorem · harmonic analysis

VectorFourier.hasFTaylorSeriesUpTo_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 : WithTop ℕ∞},
  (∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) μ) →
    MeasureTheory.AEStronglyMeasurable f μ →
      HasFTaylorSeriesUpTo N (VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L) f)
        fun w n =>
        VectorFourier.fourierIntegral Real.fourierChar μ (ContinuousLinearMap.toLinearMap₁₂ L)
          (fun v => VectorFourier.fourierPowSMulRight L f v n) w

Defined in

Mathlib.Analysis.Fourier.FourierTransformDeriv

Cited by

1 results in Mathlib

Foundations

Depth 265 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceSecondCountableTopology