I'm looking for a function $\phi$ that behaves like a standard mollifier, in particular it should be positive, even, $\phi(0)=1$, $\phi(x)=0$ for $|x|>1$ and $C^\infty$. The (scaled) mollifier \begin{equation} \epsilon(x)=e^{1-1/(1-x^2)} \end{equation} would do the job but the problem is that I'm looking for a function defined on all $\mathbb{R}$ by a single formula.

The standard mollifier is defined by cases (i.e. it's set to zero for $|x|>1$).

I think there is no such formula. Any ideas? Such function should resemble a "thin" gaussian \begin{equation} \phi(x)=e^{-x^2/\sigma} \end{equation} for very small $\sigma$; this function is anyway no good because it's not exactly zero outside the unit circle...

  • $\begingroup$ What do you mean by formula? It shouldn't be hard to concoct a formula for a function $h(x)$ that is $1$ exactly when $|x| < 1$ and $0$ otherwise. Then you could take $\epsilon(x) h(x)$. $\endgroup$ – Umberto P. Feb 2 '15 at 13:42
  • $\begingroup$ I would need a function which is not defined by cases...Actually I think it's not possible, since such a function should have a continuum of zeros, and I guess it's not possible (unless we define the function to be exactly zero in some interval) $\endgroup$ – marco trevi Feb 2 '15 at 13:53

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