Patterns That Eventually Fail
20180922T20:34:29Z  / Hacker News
We now know the two functions cross somewhere near but we don’t know if this is the first crossing!
In the original pulse, the point (0,1/2) lies on a plateau, a perfectly constant segment with a halfwidth of 1.
The process of repeatedly taking the moving average will nibble away at this plateau, shrinking its halfwidth by the halfwidth of the averaging window.
Because this is how Hanspeter Schmid explained the infamous Borwein integrals: ∫sin(t)/t dt = π/2∫sin(t/3)/(t/3) × sin(t)/t dt = π/2∫sin(t/5)/(t/5) × sin(t/3)/(t/3) × sin(t)/t dt = π/2 … ∫sin(t/13)/(t/13) × … × sin(t/3)/(t/3) × sin(t)/t dt = π/2 But then the pattern is broken: ∫sin(t/15)/(t/15) × … × sin(t/3)/(t/3) × sin(t)/t dt < π/2 Here these integrals are from t=0 to t=∞.
And Schmid came up with an even more persistent pattern of his own: ∫2 cos(t) sin(t)/t dt = π/2∫2 cos(t) sin(t/3)/(t/3) × sin(t)/t dt = π/2∫2 cos(t) sin(t/5)/(t/5) × sin(t/3)/(t/3) × sin(t)/t dt = π/2…∫2 cos(t) sin(t/111)/(t/111) × … × sin(t/3)/(t/3) × sin(t)/t dt = π/2 But: ∫2 cos(t) sin(t/113)/(t/113) × … × sin(t/3)/(t/3) × sin(t)/t dt < π/2 The first set of integrals, due to Borwein, correspond to taking the Fourier transforms of our sequence of eversmoother pulses and then evaluating F(0).
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