For real valued $x$, and $a>0$, I'm looking for an (upper) envelope of the function $$f(x)=\frac{\sin(a \pi x)}{\pi x},$$ such that the property $f(0)=a$ is still preserved.
The function
$$g(x)=\frac{1}{\pi |x|},$$
is an upper envelope of $f(x)$, but it doesn't fit at the origin. It would be helpful when the envelope is $C^1$ and as simple as possible.
There is the following possibility, although it is not particularly elegant:
Let $f(x)$ be the sigmoid-like function $$f(x)=\left\{\begin{matrix}-1 & a \pi x<-\pi/2 \\ \sin(a \pi x) & -\pi/2\leq a \pi x \leq \pi/2 \\ 1 & a \pi x > \pi/2\end{matrix}\right.$$ then $$\frac{f(x)}{\pi x}$$ is an upper envelope which fits at the origin.