TITLE: Understanding the Convolution and smoothing
QUESTION [0 upvotes]: here my question is what is mean by $f^{\epsilon}:=\eta_{\epsilon}*f$ in $U_{\epsilon}$
and how can we change form $U$ to $B(0,\epsilon)$
in the molification definition and what is use convolution in sobolev spaces  
and how can we prove that $\int \eta(x)\,dx = 1$
is that convulutions is well defined

REPLY [1 votes]: (1) The sign $:=$ means definition.
(2) In the left integral $f(y)$ only makes sense for $y\in U$. In the right integral, $\eta_\epsilon(y) = 0$ when $y\not\in B(0,\epsilon)$. For the equality see Proving commutativity of convolution $(f \ast g)(x) = (g \ast f)(x)$.
(3) What means "in the molification definition and what is use convolution in sobolev spaces"?
(4) We choose the constant $C$ for making $\int_{\Bbb R^n}\eta = 1$.
(5) As $f$ is locally integrable and $\eta_\epsilon$ has compact support the integrand (product of...) is integrable (in both integrals).
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