1. First one proves half of the claim: that is, with only
. Given
, an entire function $
$ is easily found of the form $
$ with a convenient increasing sequence of even positive integers $n_k$.
2. Then one treats the case with $f=0 < g$, that is settled with a $
$ of the form $
$ (warning: $
$ wouldn't work, for it may have poles.) This in particular gives positive real entire convolution kernels with any prescribed decay.
3. Case of $
$ A correseponding $\phi$ is a mollification of $h$ with an entire convolution kernel $
$ : $
$, which is still an entire function.
4. If $
$ is any continuous function, one writes $
$
with $
$, a series totally convergent on compacta: $
$ for all $
$.