User:PMajer/sb

1. First one proves half of the claim: that is, with only $f(x)$. Given $f$, an entire function $$\phi >f$$ is easily found of the form $$\phi (x):=\sum _{k=1}^{\infty }\left({\frac {x}{n}}\right)^{n_{k}}$$ 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 $$\phi$$ of the form $$\exp(-\psi )$$ (warning: $$1/\phi$$ wouldn't work, for it may have poles.) This in particular gives positive real entire convolution kernels with any prescribed decay. 3. Case of $$h:=(f+g)/2\in C_{c}^{0}(\mathbb {R} ).$$ A correseponding $\phi$ is a mollification of $h$ with an entire convolution kernel $$\kappa$$ : $$\phi :=h*\kappa$$, which is still an entire function. 4. If $$h\in C^{0}(\mathbb {R} )$$ is any continuous function, one writes $$h:=\sum _{j=1}^{\infty }h_{j}$$ with $$h_{j}\in C_{c}^{0}(\mathbb {R} )$$, a series totally convergent on compacta: $$\sum _{j=1}^{\infty }\|h_{j}\|_{\infty ,[-T,T]}<\infty$$ for all $$T$$.