= Solution
For a function smooth on finitely many closed pieces with finite one-sided derivatives, integration by parts gives its <Fourier series> coefficients, for $n\ne0$,
$$
c_n=\frac1{2\pi in}\sum_rJ_re^{-2\pi inx_r}+O(n^{-2}),
$$
where $J_r=f(x_r+)-f(x_r-)$ includes the jump of the periodic extension at the endpoint when necessary. In particular $|c_n|\le C/|n|$. Keeping frequencies $|n|\le K$, with $N\asymp K$, gives squared error $O(\sum_{n>K}n^{-2})=O(N^{-1})$.
A <best N-term approximation> cannot improve this worst-case rate. For example, $f=\chi_{[0,1/2)}$ has $|c_n|=1/(\pi|n|)$ on odd nonzero $n$ and zero on even nonzero $n$. Even the optimal selection therefore leaves a squared tail comparable to $N^{-1}$. Thus for the class with jumps,
$$
\boxed{\text{linear and nonlinear squared errors have sharp worst-case rate }N^{-1},}
$$
or $N^{-1/2}$ in the $L^2$ <norm>. Individual functions with no jumps, or additional cancellation, can converge faster. “Smooth except at finitely many points” must include controlled one-sided smoothness; smoothness merely on open pieces permits pathological behavior near their endpoints.
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