Researchers thought this was a bug (Borwein integrals) [video]
youtube.com
youtube.com
It shows a sequence of integrals following a very simple pattern. The first seven integrals in the sequence all evaluate to pi. The eighth integral inexplicably evaluates to pi - 0.0000000000462... and from that point on the pattern deviates from pi.
The video goes on to explain how such a seemingly perfect pattern can suddenly break by relating this sequence of integrands to a simpler one where it's easier to see what's happening.
I guess what you're saying here is that if I pass a square wave through a lowpass filter - reducing the amplitude of the harmonics and rounding off the corners - then the peak amplitude will stay pretty much constant until I pull the cutoff of the filter down sufficiently close to the fundamental that it starts getting attenuated too.
Makes sense I guess.
I suppose: in the original problem we want to see what a sinc multiplied by sinc looks like (in time domain) or rect convolved with rect (analyzing in fourier domain). Also the width of the rects we're convolving with is shrinking after each time.
As you mentioned, to see what a rect convovled with a rect looks like, I think you can treat a convolution with rect as a lowpass filter (this should not be confused with convolving with sinc which gives an ideal" lowpass filter), and this gives intuition for why the erosion occurs.
What's not clear a priori to me is that the erosion will indeed actually reach the center. I think this depends on how fast the rect you're convolving with is shrinking, if it shrunk faster than {1, 1/3, 1/5, ...} then I don't think it would. I guess the easiest way to see this visually is to use the sliding method Grant showed, where the width of the y=1 peak after convolution is the overlapping width of the two rects. Thus we get 1, 1-1/3, 1-1/3-1/5,... which eventually < 0
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.
I sometimes regret not studying pure math in college, and going down the software engineer (ahem, code monkey) route. There's so much mathematical beauty out there to be discovered and admired.
But I guess money's better this way.