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.