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 $$$h:=(f+g)/2\in C^0_c(\mathbb{R}).$$$ 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 $$$h_j\in C^0_c(\mathbb{R})$$$, a series totally convergent on compacta: $$$\sum_{j=1}^\infty \|h_j\|_{\infty,[-T,T]}<\infty$$$ for all $$$T$$$.