I'm in need of a well studied bell curve function $f(x)$ that satisfies the following equation:
$\exists P,\!C \; \forall x \:\left(\sum_{i=-\infty}^{\infty}{f(x+i P)}\right)=C$
That is; an infinite set of them offset by some fixed interval sums to the same value everywhere.
[[NOTE: if this is a well studied field, I seem to be unable to figure what terms to lookup to do my own research. I'd welcome just being pointed at a search term I can use to answer this on my own. That would likely be more valuable than an answer to the question I'm actually asking here.]]
Ideally I'd like a function that:
- Has a maximum at zero.
- Is monotonic going away from zero.
- Is positive everywhere.
- Is continuous and continuously differentiable.
- Is symmetric around zero.
I'm looking to do interpenetration from periodic samples, but wants something that has different properties than a Taylor series (no divergence to infinity) or a discreet Fourier transform (an assumption of being periodic). Also I need it to be highly local: altering a sample value should have rapidly diminishing effects away from that value.
Off hand the only thing I'm coming up with is this form of function:
$f(x) = tan^{-1}(x+\frac{P}{2}) - tan^{-1}(x-\frac{P}{2})$
The same thing using a logistics curve or some other function with similar shape would also work.