l = 0,1, \ldots$$ then $$\label{Eg-leq-nk1}
E_n^T(g) \, \leq\, \frac{c^{l+1} M_{l+1}} {n^{l+1}}, ~~~
c = 1 + \frac{\pi^2}{2}$$ For reasons that will become apparent in Section \[sec:Interpolation-argument\], we are interested primarily in trigonometric *interpolation* rather than the best approximation, and thus need to relate the two. The interpolant $\tilde{T}_N(\phi)$ of a given function $g(\phi)$ over a set of $N = 2n + 1$ equidistant knots $\{\phi_m \}$ is defined in a standard way, by requiring that $$\label{eqn:interpolant-defined}
   \tilde{T}_N(g, \phi_m) = g(\phi_m), ~~ \phi_m = \frac{2\pi m}{N},
   ~~~m = 0,1, \ldots, N-1$$ It is known that this interpolant exists and is unique. Furthermore, there is an upper bound for the interpolation error: $$\label{eqn:interp-error-vs-approximation-error}
   \| g - \tilde{T}_N(g) \|_\infty ~ \leq ~ (1 + \Lambda_N) \,
   \| g - T_N \|_\infty ~\equiv~ (1 + \Lambda_N) \, E_n^T(g)$$ where $\Lambda_N$ is the Lebesgue constant, which itself has an upper bound [@Cheney1975] $$\label{eqn:Lebesgue-const-upper-bound}
\Lambda_N \, \leq \, 2 \pi^{-1} \log N + \frac{5}{3}$$ All the above information can be found in a variety of sources, including very recent ones [@Austin-Trefethen-trig-interp2017; @Austin-PhD-thesis-interpolation2016], [@Trefethen-Weideman-exp-convergent-trapezoidal2014 Section 7].

Combining , , and , one has $$\label{eqn:interp-error-algebraic-convergence-vs-N}
\| g - \tilde{T}_N(g) \| \, \leq \, \left(2 \pi^{-1} \log N 