$f$ can be analytically continued to a $2\pi$-periodic function for $-\delta < \mathrm{Im} \, x < \delta$ for some $\delta > 0$, then for any $\hat \delta < \delta$, $$\label{eqn:Austin-Trefethen-trig-interp-accuracy-exp}
| I - \tilde I_N^{} | , \| f - \tilde t_N^{} \| = 
\mathcal{O}(\exp(-\hat \delta N)).$$ 0.1in

The qualitative conclusion of this section is that *trigonometric interpolation of a smooth periodic function provides a very accurate approximation of the function and its integrals.*

Preliminaries: Finite Difference Trefftz Schemes {#sec:FD-Trefftz}
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Another preliminary subject, which will be needed in [Section \[sec:FLAME-slab\]]{}, is FLAME [@Tsukerman-JCP10; @Tsukerman05; @Tsukerman06; @Tsukerman-PBG08; @Tsukerman-book07; @Pinheiro07]. Recall that classical FD schemes are typically derived from Taylor expansions; but this is problematic if the solution is not sufficiently smooth – e.g. at material interfaces. That is the root cause of the notorious “staircase” effect at slanted or curved interface boundaries that do not conform geometrically to the grid lines. FLAME replaces Taylor polynomials with Trefftz functions, which often produces high-order schemes.

The key ideas of FLAME are as follows. Let a boundary value problem be defined in a computational domain $\Omega$ and consider a small subdomain $\Omega_h$ within which a difference scheme is to be formed. In $\Omega_h$, 