The (open) patches $\omega_i$, $i=0,\ldots,N$, are given by the supports of the functions $\varphi_i$, i.e., $\omega_i = (\operatorname*{supp} \varphi_i)^\circ
 = K_i \cup K_{i+1} \cup \{x_i\}$.

Polynomial approximation (see, e.g., [@MelenkCLEM Proposition [A.2]{}]) gives the existence of an interpolation operator $J_p:L^2(-1,1) \rightarrow {\mathbb P}_p(-1,1)$ that is uniformly (in $p\ge 0$) stable, i.e., $\|J_p v\|_{L^2(-1,1)} \leq C \|v\|_{L^2(-1,1)}$ for all $v \in L^2(-1,1)$ and has the following properties for $v \in H^1(-1,1)$: $$(p+1) \|v - J_p v\|_{L^2(-1,1)} + 
\|(v - J_p v)^\prime\|_{L^2(-1,1)} \leq  C \|v^\prime\|_{L^2(-1,1)}.$$ Furthermore, if $v$ is antisymmetric with respect to the midpoint $x = 0$, then $J_p v$ can be assumed to be antisymmetric as well, i.e., $(J_p v)(0) = 0$ (this follows from studying the antisymmetric part of the original function $J_p v$).

The approximation ${\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v$ is now constructed with the aid of a “partition of unity argument” as described in [@babuska-melenk96 Theorem [2.1]{}]. For $\omega_0$ and $\omega_N$, extend $v$ anti-symmetrically, i.e., $v(x):=-v(-x)$ for $x \in K_{0}$ and $v(x):=-v(1-x)$ for $x \in K_{N+1}$. Then $v$ is defined on each patch $\omega_i$, $i=0,\ldots,N$. For each patch $\omega_i$, let $p^\prime_i:= \min\{p_{i},p_{i+1}\}$ (with the understanding $p_0 = p_1$ and $p_{N+1} = p_N$). The above operator $J_p$ then induces for 