Theorems · Theorem · integral transforms

HasCompactSupport.hasFDerivAt_convolution_right

∀ {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'} [inst_3 : RCLike 𝕜]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace ℝ F] [inst_7 : NormedSpace 𝕜 F]
  [inst_8 : MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [inst_9 : NormedAddCommGroup G]
  [BorelSpace G] [inst_11 : NormedSpace 𝕜 G] [MeasureTheory.SFinite μ] [μ.IsAddLeftInvariant],
  HasCompactSupport g →
    MeasureTheory.LocallyIntegrable f μ →
      ContDiff 𝕜 1 g →
        ∀ (x₀ : G),
          HasFDerivAt (MeasureTheory.convolution f g L μ)
            (MeasureTheory.convolution f (fderiv 𝕜 g) (ContinuousLinearMap.precompR G L) μ x₀) x₀

Compute the total derivative of f ⋆ g if g is C^1 with compact support and f is locally integrable. To write down the total derivative as a convolution, we use ContinuousLinearMap.precompR.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Convolution

Cited by

2 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupRCLikeNormedSpaceNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedAddCommGroupBorelSpaceNormedSpaceMeasureTheory.SFiniteMeasureTheory.Measure.IsAddLeftInvariant