Construction of specific test function I want to construct a test function with value 1 on the unit ball which vanishes für $\vert x\vert\geq 2$. I tried to do so by performing a convolution of the function $c\cdot\exp(1/(x^2-1))$ (c a normalizing constan) with an indicator function $1_{\lbrack-a,a\rbrack}$, but this isn't constant for $\vert x\vert\leq1$. I hope you can help me :)
 A: One standard technique for doing this is to use a radial function that decays to zero using a rescaled and shifted version of the behavior of $e^{-1/x}$ for $x > 0$ as $x \rightarrow 0^{+}$. You can then define your test function as a function of $|x|$ piecewise, with three pieces; one for $|x| \le 1$, another for $|x| \ge 2$, and the rapidly decaying portion on $1 < |x| < 2$.
A: You should probably had checked why it doesn't fulfill your requirements. Convolution with a function with unit intagral with support in a origin centered ball of radius $r$ will result that $\inf f(D_a)<\phi(a)<\sup f(D_a)$ where $D_a$ is the ball around $a$. This means that if $f(D_a)$ only has one value then $\phi(a)$ will take that value.
So for an indicator function the convolution will take the value $0$ at distance $r$ or greater from the set, and the value at distance and $1$ at distance $r$ or greater from the complement.
So you want it to take $1$ inside the unit ball and $0$ outside the double-unit ball? Just take a set that has both positive distance to the outer region and whose complement has positive distance to the unit ball. The $3/2$-unit ball is such a set with distances $1/2$. So you use that and a test function with unit integral and support in the ball with radius $1/2$ around the origin.
