Theorems · Theorem · abstract harmonic analysis

MeasureTheory.convolution_eq_swap

∀ {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup E'] [inst_2 : NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G}
  [inst_3 : NontriviallyNormedField 𝕜] [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace 𝕜 F]
  (L : E →L[𝕜] E' →L[𝕜] F) [inst_7 : MeasurableSpace G] {μ : MeasureTheory.Measure G} [inst_8 : NormedSpace ℝ F]
  [inst_9 : AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G],
  MeasureTheory.convolution f g L μ x = ∫ (t : G), (L (f (x - t))) (g t) ∂μ

The symmetric definition of convolution.

Defined in

Mathlib.Analysis.Convolution

Cited by

3 results in Mathlib

Foundations

Depth 259 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedSpaceAddCommGroupMeasureTheory.Measure.IsAddLeftInvariantMeasureTheory.Measure.IsNegInvariantMeasurableNegMeasurableAdd