Any curve can be represented as a sum of sinusoidal curves, i.e. a Fourier series.
Not quite. Any
periodic curve can be represented as a sum of sinusoidal curves. The trick is in knowing the periodicity of $=f(t). protip: that f() ain't periodic.
Nope.
An infinite fourier series can match any continuous differentiable function.
An infinite series of a non-deterministic function is unrealizable.
Unless you know the function a priori, you cannot derive a convolution thereof. If you know the function a priori, there is no need to convolve it in order to profit. The entire concept of obtaining a Fourier series of $=f(t) is therefore nonsensical.
Obviously we don't know what the price curve is going to be in the future. That's not the point. The point is that if the price starts at f(t_0) = X and ends at the same price f(t_1) = X, and in between it has lots of ups and downs, then this method will increase the end-value of your total holdings (expressed in BTC or in $). It doesn't matter what f(t) does in between. You don't have to know ahead of time what f(t) is going to do. The demonstration of this statement makes use of the fact that f(t) = the sum of lots of sine waves, for
any f(t). This is true even if we don't know ahead of time what exactly f(t) is.