[proofplan] We combine the exponential coefficient bound $|\widehat{h}_n| \leq Mc^{|n|}$ (from the [Exponential Convergence of Fourier Approximation](/theorems/1379)) with the [Aliasing Formula for the Rectangle Rule](/theorems/1381). Applying the coefficient bound to each aliased mode $\widehat{h}_{Nr}$ with $|r| \geq 1$ and summing the resulting geometric series yields the exponential convergence estimate. [/proofplan] [step:Apply the coefficient decay bound to the aliasing formula] By the [Aliasing Formula for the Rectangle Rule](/theorems/1381), the quadrature error is \begin{align*} I_N(h) - I(h) = 2\sum_{\substack{r \in \mathbb{Z} \\ |r| \geq 1}} \widehat{h}_{Nr}. \end{align*} Taking absolute values and applying the triangle inequality: \begin{align*} |I_N(h) - I(h)| \leq 2\sum_{\substack{r \in \mathbb{Z} \\ |r| \geq 1}} |\widehat{h}_{Nr}|. \end{align*} Since $h$ is 2-periodic and analytic on the strip $\{z : |\operatorname{Im} z| < a\}$ with $|h(z)| \leq M$, the [Exponential Convergence of Fourier Approximation](/theorems/1379) (part 1) gives $|\widehat{h}_n| \leq Mc^{|n|}$ for all $n \in \mathbb{Z}$, where $c = e^{-a\pi}$. Applying this with $n = Nr$ (so $|n| = N|r|$): \begin{align*} |\widehat{h}_{Nr}| \leq M c^{N|r|}. \end{align*} [/step] [step:Sum the geometric series to obtain the exponential bound] Substituting the coefficient bound: \begin{align*} |I_N(h) - I(h)| \leq 2\sum_{|r| \geq 1} Mc^{N|r|} = 2M \cdot 2\sum_{r=1}^{\infty} c^{Nr} = 4M\sum_{r=1}^{\infty} (c^N)^r, \end{align*} where we grouped positive and negative values of $r$ (each contributing the same amount $Mc^{Nr}$) and factored $c^{Nr} = (c^N)^r$. Since $c \in (0,1)$ and $N \geq 1$, we have $c^N \in (0,1)$, so the geometric series converges: \begin{align*} \sum_{r=1}^{\infty} (c^N)^r = \frac{c^N}{1 - c^N}. \end{align*} Therefore \begin{align*} |I_N(h) - I(h)| \leq \frac{4Mc^N}{1-c^N}. \end{align*} [/step]