relatively small segments of the structure ([Section \[sec:FLAME-slab\]]{}).

Sections \[sec:Trig-interpolation\] and \[sec:FD-Trefftz\] provide background information needed in the application examples of Sections \[sec:Trefftz-homogenization\] and \[sec:FLAME-slab\]. The underlying mechanisms for the accuracy of Trefftz approximations are discussed in Section \[sec:Trefftz-approximation\].

Preliminaries: Trigonometric Projection and Interpolation {#sec:Trig-interpolation}
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Trigonometric approximation of periodic functions is a well-established subject. Here we summarize the key mathematical results that will be needed in Section \[sec:Interpolation-argument\].

For any Lipschitz-continuous periodic function $g$ on $[-\pi, \pi]$, one may consider its best possible approximation by a trigonometric polynomial $T_n$ in the maximum norm: $$\label{eqn:En-best-approx-trig-poly}
E_n^T(g) \,=\, \min_{\alpha, \beta} \max_{\phi \in [-\pi, \pi]}
\left| g(\phi) - T_n(\alpha, \beta, \phi) \right|, ~~~$$ where $$\label{eqn:trig-poly-defined}
T_n(\alpha, \beta, \phi) \, \equiv \, \alpha_0 + \sum_{\nu=1}^n 
(\alpha_\nu \cos \nu \phi + \beta_\nu \sin \nu \phi),$$ $$\alpha \equiv \{\alpha_0, \alpha_1, \ldots, \alpha_n\},
~~~ \beta \equiv \{\beta_1, \ldots, \beta_n\}$$ A slightly modified notation of [@Meinardus-Approximation-theory1967] is used here. Note that the total number of coefficients $\alpha$, $\beta$ in the trigonometric series is $N = 2n+1$.

It follows from Jackson’s theorem [@Jackson-Theory-approximation1930], or [@Meinardus-Approximation-theory1967 Theorem 41], that if the derivative $g^{(l+1)} (\phi)$ exists and is bounded, i.e. $$\label{g-bounded-derivative}
\left| g^{(l+1)} (\phi) \right|  \, \leq \, M_{l+1},
~~~ 