Theorems · Theorem · harmonic analysis

VectorFourier.fourierIntegral_comp_add_right

∀ {𝕜 : Type u_1} [inst : CommRing 𝕜] {V : Type u_2} [inst_1 : AddCommGroup V] [inst_2 : Module 𝕜 V]
  [inst_3 : MeasurableSpace V] {W : Type u_3} [inst_4 : AddCommGroup W] [inst_5 : Module 𝕜 W] {E : Type u_4}
  [inst_6 : NormedAddCommGroup E] [inst_7 : NormedSpace ℂ E] [MeasurableAdd V] (e : AddChar 𝕜 Circle)
  (μ : MeasureTheory.Measure V) [μ.IsAddRightInvariant] (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (v₀ : V),
  VectorFourier.fourierIntegral e μ L (f ∘ fun v => v + v₀) = fun w =>
    e ((L v₀) w) • VectorFourier.fourierIntegral e μ L f w

The Fourier integral converts right-translation into scalar multiplication by a phase factor.

Defined in

Mathlib.Analysis.Fourier.FourierTransform

Cited by

1 results in Mathlib

Foundations

Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleMeasurableSpaceAddCommGroupModuleNormedAddCommGroupNormedSpaceMeasurableAddMeasureTheory.Measure.IsAddRightInvariant