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regularize a function (proof function)


I want to regularize a function, I want that it is C^inf and with
compact support (proof function for distributions).

I start with:

a = -1; b = 1;
theta[x_] := (-x^2) + 1
f[x_] := If[x < b && x > a, theta[x], 0]
Plot[f[x], {x, -2, 2}];

Now f[x] is with compact support, but it's not C^inf. I do:

phi[epsilon_, x_] := If[x >=epsilon || x
    <= (-epsilon), 0, Exp[(-epsilon^2)/((epsilon^2 - x^2))]])

Plot[phi[1,x],{x,-1,1}];

this is a proof function. I define
g[epsilon_, x_] := (Int[phi[epsilon, Abs[x - y]] f[y] {y,-inf,inf}]) /
(Int[phi[epsilon, Abs[y]] {y,-inf,inf}])

Now I want to see it:
Plot[g[1,x],{x,-2,2}]

It has a loss of precision, how can I solve it?


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