Theorems · Theorem · integral transforms

HasCompactSupport.hasFDerivAt_convolution_left

∀ {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] [μ.IsNegInvariant],
  HasCompactSupport f →
    ContDiff 𝕜 1 f →
      MeasureTheory.LocallyIntegrable g μ →
        ∀ (x₀ : G),
          HasFDerivAt (MeasureTheory.convolution f g L μ)
            (MeasureTheory.convolution (fderiv 𝕜 f) g (ContinuousLinearMap.precompL G L) μ x₀) x₀

Defined in

Mathlib.Analysis.Calculus.ContDiff.Convolution

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupRCLikeNormedSpaceNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedAddCommGroupBorelSpaceNormedSpaceMeasureTheory.SFiniteMeasureTheory.Measure.IsAddLeftInvariantMeasureTheory.Measure.IsNegInvariant