Theorems · Theorem · integral transforms

HasCompactSupport.hasDerivAt_convolution_left

∀ {𝕜 : Type u𝕜} {E : Type uE} {E' : Type uE'} {F : Type uF} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup E'] [inst_2 : NormedAddCommGroup F] [inst_3 : RCLike 𝕜] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace ℝ F] [inst_7 : NormedSpace 𝕜 F] {f₀ : 𝕜 → E} {g₀ : 𝕜 → E'}
  (L : E →L[𝕜] E' →L[𝕜] F) {μ : MeasureTheory.Measure 𝕜} [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ]
  [μ.IsNegInvariant],
  HasCompactSupport f₀ →
    ContDiff 𝕜 1 f₀ →
      MeasureTheory.LocallyIntegrable g₀ μ →
        ∀ (x₀ : 𝕜), HasDerivAt (MeasureTheory.convolution f₀ g₀ L μ) (MeasureTheory.convolution (deriv f₀) g₀ L μ x₀) x₀

Defined in

Mathlib.Analysis.Calculus.ContDiff.Convolution

Cited by

0 results in Mathlib

Foundations

Depth 266 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupRCLikeNormedSpaceNormedSpaceNormedSpaceNormedSpaceMeasureTheory.Measure.IsAddLeftInvariantMeasureTheory.SFiniteMeasureTheory.Measure.IsNegInvariant