Dacyn comments on Paper: “A simple function of one parameter (θ) can fit any collection of ordered pairs {Xi,Yi} to arbitrary precision”—implications for Occam?

Dacyn 31 May 2018 20:11 UTC
6 points
They want all of the error to be in the vertical direction. So e.g. sin(x/T) cannot approximate (0,1) in their sense for any T. Similarly, it cannot simultaneously approximate (1,0) and (2,1).
- paulfchristiano 1 Jun 2018 7:33 UTC
  0 points
  Parent
  Sorry, we need sin(ax + b).
  - Dacyn 1 Jun 2018 8:39 UTC
    4 points
    Parent
    Still doesn’t work. If $(x_{i})_{i = 1}^{3}$ is an arithmetic progression then $(exp (i (a x_{i} + b)))_{i = 1}^{3}$ is a geometric progression, the imaginary part of which is $(sin (a x_{i} + b))_{i = 1}^{3}$ . This implies that for example $sin (a x + b)$ cannot simultaneously approximate $(0, 0)$ , $(1, 0)$ , and $(2, 1)$ .