4 ms·

"if one thinks of functions as fuzzy versions of points" Can anyone elaborate on this?

• peppery 6y ago
  Tao is likely inviting us (just as many physical/probabilistic laws do) to view any arbitrary function as relatively "thicker"/"fuzzier" than an infinitely-thin, infinitely-tall spike function at a certain value: the Dirac delta function (https://en.wikipedia.org/wiki/Dirac_delta_function https://en.wikipedia.org/wiki/Dirac_delta_function). If you convolve ≡ integrate this Dirac delta function (located at some value x) against any function g(t), by construction the integral is zero everywhere except at t=x, so the result is an infinitely thin slice of g at x, exactly g(x) (the 'sifting property,' https://math.stackexchange.com/questions/1015498/convolution-with-delta-function https://math.stackexchange.com/questions/1015498/convolution...). Now imagine you begin to thicken/fuzz the spike; now you begin to accumulate the behavior of g(t) not just exactly at x, but also at points nearby, getting a schmeared representation of g. Deforming our spike to an arbitrary function of interest in this way gives an arbitrary convolution schmear.
  • jonahx 6y ago
    Great explanation, thank you.