= Solution
The exact average is $I(h)=\widehat h_0$. For one Fourier mode,
$$
\frac1{2N}\sum_{k=-N+1}^Ne^{i\pi nk/N}
=\begin{cases}
1,&2N\mid n,\\
0,&2N\nmid n.
\end{cases}
$$
Thus the <periodic trapezoidal Fourier aliasing> identity gives
$$
I_N(h)=\sum_{j\in\mathbb Z}\widehat h_{2Nj},
\qquad
\boxed{|I_N(h)-I(h)|
=\left|\sum_{j\ne0}\widehat h_{2Nj}\right|}.
$$
Under the coefficient bound,
$$
|I_N-I|
\leq2M\sum_{j=1}^\infty c^{2Nj}
=\boxed{\frac{2Mc^{2N}}{1-c^{2N}}}
\leq\frac{2M}{1-c^2}c^{2N}.
$$
This decays exponentially in $N$.
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