I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S)
how the heck is that a fourier transform!??
What you described is more often referred to as an “indicator function” these days, with “characteristic functions” denoting the transform (Fourier, laplace, z - depending on context). Closely related to “moment generating functions” to the point of being almost interchangeable.
but the new and improved 21st century characteristic functions are n-variable and have a full continious spectrum of variables between zero (false) and one (true) but only potentially lest infinite realizes itself (which would make the theories illogical).
this way of thinking about this makes sense to me, even if it's ever so slighly wrong by some nitpickable point https://en.wikipedia.org/wiki/Moment-generating_function
- The characteristic function of a random variable X is defined as the function that maps t --> ExpectedValue[ exp( i * t * X ) ]
- Computing this expected value is the same as regarding t as a constant and integrating the function x --> exp( i * t * x) with respect to the distribution of X, i.e. if X has the density f, we compute the integral of f(x) * exp( i * t * x) with respect to x over the domain of f.
- on the other hand: computing the Fourier transform of f (here representing the density of X) and evaluating it at point t (i.e. computing (F(f))(t) if F represents the Fourier transform) is the same as fixing t and computing the integral of f(x) * exp( -i * t * x) with respect to x.
- Rearranging the integrand in the previous expression to f(x) * exp( i * -t * x), we see that it is the same as the integrand used in the characteristic function, only with a -t instead of a t.
Hope that helps :)