= Solution
Insert the <Fourier series>
$$
h(t)=\sum_{n\in\mathbb Z}\widehat h_n e^{i\pi nt}
$$
into the rectangle rule. The finite geometric sum obeys
$$
\frac1N\sum_{k=-N/2+1}^{N/2}
e^{2\pi ink/N}
=
\begin{cases}
1,&N\mid n,\\
0,&N\nmid n.
\end{cases}
$$
Consequently the quadrature value is
$$
\frac2N\sum_{k=-N/2+1}^{N/2}h(2k/N)
=2\sum_{j\in\mathbb Z}\widehat h_{jN},
$$
whereas
$$
\int_{-1}^1h(t)\,dt=2\widehat h_0.
$$
Thus the exact <Periodic N-point rectangle-rule error> is
$$
\boxed{
e_N(h)=-2\sum_{\substack{j\in\mathbb Z\\j\ne0}}
\widehat h_{jN}.}
$$
By the <Fourier coefficient decay of an analytic periodic function>, there are constants $C>0$ and $0<\rho<1$ such that
$$
|\widehat h_n|\leq C\rho^{|n|}.
$$
It follows that
$$
|e_N(h)|
\leq4C\sum_{j=1}^{\infty}\rho^{jN}
=\frac{4C\rho^N}{1-\rho^N}.
$$
Therefore
$$
\boxed{|e_N(h)|=O(\rho^N),}
$$
which is exponential, and hence spectral, convergence.
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