Distributions and Sobolev Spaces

Abstract

We shall begin with some elementary results concerning the approximation of functions by very smooth functions. For each ε > 0, let ϕ ε ∈ C ∞ 0 (R n) be given with the properties ϕ ε ≥ 0 , supp(ϕ ε) ⊂ {x ∈ R n : |x| ≤ ε} , ϕ ε = 1. Such functions are called mollifiers and can be constructed, for example, by taking an appropriate multiple of ψ ε (x) = exp(|x| 2 − ε 2) −1 , |x| < ε , 0 , |x| ≥ ε .