+ \frac{8}{3} \right) 
\left(1 + \frac{\pi^2}{2} \right)^{l+1} \frac{M_{l+1}}  {n^{l+1}}$$ This indicates fast uniform algebraic convergence of the interpolant with respect to the number of knots. Moreover, under additional assumptions of analyticity of $g(\theta)$ in a strip of the complex plane Re$\, \theta \in (0, 2\pi)$, $|\mathrm{Im} \, \theta | < \delta$, convergence becomes exponential [@Trefethen-Weideman-exp-convergent-trapezoidal2014 (7.19)]: $$\label{eqn:interp-error-exp-convergence-vs-N}
\| g - \tilde{T}_N(g) \| \, \leq \, \frac{4M \exp[-\delta(N+1)/2]}
{1 - \exp(-\delta)}$$ We are also interested in the approximation of the integral $$\label{eqn:integral-of-g}
I \,=\, \int_0^{2\pi} g(\theta) \, d \theta$$ using the values of $g$ at the equispaced knots: $$\label{eqn:num-quadrature-of-g}
I_N \,=\, \frac{2\pi}{N} \, \sum_{m=1}^{N-1} g(\theta_m),
~~~ \theta_m = \frac{2\pi m}{N},$$ which is the trapezoidal rule for the numerical quadrature. Under the same analyticity assumptions as above, the error of this quadrature can be bounded as [@Trefethen-Weideman-exp-convergent-trapezoidal2014 (7.20)] $$\label{eqn:quadrature-error-exp-convergence-vs-N}
\| I_N - I \| \, \leq \, \frac{8\pi M \exp[-\delta(N+1)/2]}
{1 - \exp(-\delta)}$$ A similar result can be found in [@Austin-Trefethen-trig-interp2017 Theorem 1]. Adapted to our needs and notation, it states:

0.1in If $f$ is $l$ times continuously differentiable and $f^{(l)}$ is Lipschitz continuous, then $$\label{eqn:Austin-Trefethen-trig-interp-accuracy-power}
| I - \tilde I_N^{} | ,  \| f - \tilde t_N^{} \| 
= \mathcal{O}(N^{-(l+1)}).$$ If 