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User:Quietbritishjim/Approximate identity

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From Folland's book on PDEs

Lemma

Let fL1(ℝn) be continuous.

Let φL1(ℝn) be bounded and such that φ(x) is positive and decreasing in |x| and

Set

(Note that also .) Then for each x∈ℝn

Proof

Given η>0 we need to show that there exists α>0 such that, for all 0<ε<α, we have

Since f is continuous there exists α1(η)>0 such that

where |B| is the volume of the unit ball in ℝn; in particular

It thus remains to find an α2(η)>0 (then set α:=min{α1,α2}) such that

But this expression is bounded by the sum of

so such a value for α2(η) must exist.