rect(x) • rect(x/2) • rect(x/4) • ...
Where • is the convolution operator.
Unlike the series in the video, 1 + 1/2 + 1/4 + ... converges. So this function has compact support, and the value at 0 does not dip.
I expect it to be a https://en.m.wikipedia.org/wiki/Bump_function
It might be possible to get a closed-form solution via an approach like [1]. (Out of curiosity, for rect convolved with itself, the intermediaries seem to be knwon as b-splines: https://www.chebfun.org/examples/approx/BSplineConv.html)
[1] https://math.stackexchange.com/questions/1254392/the-maximum...
You get a Gaussian by repeated convolution of the same function (and normalizing the width).
The equivalent question to what I asked is what's the pdf of X1 + X2/2 + X3/4 + ...
where Xi is iid unit uniform.