Theorems · Theorem · global analysis

ContDiffBump.convolution_eq_right

∀ {G : Type uG} {E' : Type uE'} [inst : NormedAddCommGroup E'] {g : G → E'} [inst_1 : MeasurableSpace G]
  {μ : MeasureTheory.Measure G} [inst_2 : NormedSpace ℝ E'] [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace ℝ G]
  [CompleteSpace E'] {φ : ContDiffBump 0} [inst_6 : HasContDiffBump G] {x₀ : G},
  (∀ x ∈ Metric.ball x₀ φ.rOut, g x = g x₀) →
    MeasureTheory.convolution (↑φ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀ = MeasureTheory.integral μ ↑φ • g x₀

If φ is a bump function, compute (φ ⋆ g) x₀ if g is constant on Metric.ball x₀ φ.rOut.

Defined in

Mathlib.Analysis.Calculus.BumpFunction.Convolution

Cited by

0 results in Mathlib

Foundations

Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasurableSpaceNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceHasContDiffBump