Does anyone know how to calculate the error between a function and its Fourier series representation as a function of the partial sums of the series? So far I haven't been able to find anything in the literature that talks about this. I'm also interested in looking at how well a Fourier series can approximate angles. My guess is that the error approaches a non-zero value at the angle as you take an infinite number of terms because every term in a Fourier series is smooth and continuous. I just stumbled across the Gibbs phenomenon after a Google search, so I'll be looking into that. But if anyone has anything else to add, I'd appreciate it. Thanks in advance.