Periodic N-point rectangle-rule error
= Periodic N-point rectangle-rule error
{c}
{title2=$e_N(h)$}
For a two-periodic function and $N$ equally spaced samples,
$$
\frac2N\sum_{k=-N/2+1}^{N/2}h(2k/N)
=2\sum_{j\in\mathbb Z}\widehat h_{jN}.
$$
Consequently
$$
e_N(h)=-2\sum_{j\ne0}\widehat h_{jN}.
$$
Exponential <Fourier coefficient> decay therefore gives exponential convergence in $N$.