Theorems · Theorem · global analysis

ContDiffBump.normed_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} [BorelSpace G] [inst_7 : FiniteDimensional ℝ G]
  [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] {x₀ : G},
  (∀ x ∈ Metric.ball x₀ φ.rOut, g x = g x₀) →
    MeasureTheory.convolution (φ.normed μ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀ = g x₀

If φ is a normed bump function, compute φ ⋆ g 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 283 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupMeasurableSpaceNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceBorelSpaceFiniteDimensionalMeasureTheory.IsLocallyFiniteMeasureMeasureTheory.Measure.IsOpenPosMeasure