I wonder what the Fourier transform of the Mandelbrot set looks like? More specifically, the 2D Fourier transform of the function f(x,y) = {0 if x+iy is in M, 1 otherwise}. This has infinitely fine features, so the Fourier transform will extend out infinitely far from the origin. It’s aperiodic, so the Fourier transform will be non-discrete.
The result will be a complex-valued function of complex numbers (since each point in the frequency domain has a phase and amplitude). That raises the question of its analytical properties - is it analytic everywhere, in some places or nowhere? (Probably nowhere).
Other interesting Mandelbrot-set related functions that could also be Fourier transformed:
M_n(x,y) = the nth iterate of the Mandelbrot equation (f = |exp(-(lim(n->infinity) M_n)/n)|).
D(x,y) = distance between x+iy and the closest point in the Mandelbrot set. Phase could also encode direction.
P(x,y) = the potential field around an electrically charged Mandelbrot set.