with respect to the local element sizes $h_j$ and polynomial degrees $p_j$, it is crucial to construct an interpolation operator that is *simultaneously* $L^2$- and $H^1$-stable. This will be accomplished in the current section; see Proposition \[pr:intp\] and Corollary \[cr:int\] below.

\[pr:intp\] Let the pair $({\mathcal{T}},{\bm p})$ be $\mu$-shape regular; see . Then, there exists an interpolation operator ${\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}:\,H^1_0(\Omega)\to{V^{\rm hp}({\mathcal{T}},{\bm p})}$ such that, for any $j=1,2,\ldots,N$, and for all $v\in H^1_0(\Omega)$, there holds $$\label{eq:CI}
\begin{split}
{\left\|v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v\right\|}_{L^2(K_j)}&\le C_I\min\left\{{\left\|v\right\|}_{L^2(\widetilde K_j)}, h_jp_j^{-1}{\left\|v'\right\|}_{L^2(\widetilde K_j)}\right\},\\
{\left\|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)'\right\|}_{L^2(K_j)}&\le C_I{\left\|v'\right\|}_{L^2(\widetilde K_j)}.
\end{split}$$ Here, $C_I>0$ is a constant that depends solely on $\mu$; in particular, it is independent of $v$, $\mathcal{T}$, and of $\bm p$.

Let us, without loss of generality, assume that $\Omega=(0,1)$. The result can be shown with the techniques employed in the higher-dimensional case in [@karkulik-melenk15 Cor. [3.7]{}]. In the present, one-dimensional case, a significantly simpler argument can be brought to bear, which allows us to make the paper self-contained. Let $x_{-1} = -h_1$ and $x_{N+1} = 1+h_N$ and $\varphi_i$, $i=0,\ldots,N+1$ be the standard piecewise linear hat functions associated with the nodes $x_i$, $i=-1,\ldots,N+1$. The extra nodes $x_{-1}$ and $x_{N+1}$ define in a natural way the elements $K_{0}$ and $K_{N+1}$. 