Theorems · Theorem · harmonic analysis

Real.integrable_prod_sub

∀ {𝕜 : Type u_1} {E : Type u_3} {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedAddCommGroup F₁] [inst_3 : NormedAddCommGroup F₂]
  [inst_4 : NormedAddCommGroup F₃] [inst_5 : InnerProductSpace ℝ E] [inst_6 : FiniteDimensional ℝ E]
  [inst_7 : MeasurableSpace E] [inst_8 : BorelSpace E] [inst_9 : NormedSpace 𝕜 F₁] [inst_10 : NormedSpace 𝕜 F₂]
  [inst_11 : NormedSpace 𝕜 F₃] (B : F₁ →L[𝕜] F₂ →L[𝕜] F₃) {f₁ : E → F₁} {f₂ : E → F₂},
  MeasureTheory.Integrable f₁ MeasureTheory.volume →
    MeasureTheory.Integrable f₂ MeasureTheory.volume →
      MeasureTheory.Integrable (fun p => ‖B‖ * (‖f₁ (p.1 - p.2)‖ * ‖f₂ p.2‖))
        (MeasureTheory.volume.prod MeasureTheory.volume)

The norm of the integrand of the convolution is integrable if the functions are integrable.

Defined in

Mathlib.Analysis.Fourier.Convolution

Cited by

0 results in Mathlib

Foundations

Depth 260 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceNormedSpaceNormedSpaceNormedSpace