each patch $\omega_i$ by scaling an operator $J^i:L^2(\omega_i) \rightarrow {\mathcal P}_{p^\prime_i-1}(\omega_i)$ with the following properties: $$\frac{p_i^\prime+1}{h_i}\|v - J^i v\|_{L^2(\omega_i)} + 
\|(v - J^i v)^\prime\|_{L^2(\omega_i)} \leq  C \|v^\prime\|_{L^2(\omega_i)};$$ here, we have exploited the $\mu$-shape regularity of the mesh. We note that $(J^0 v)(0) = 0$ and $(J^N v)(1) = 0$. Also, the operators $J^i$ are uniformly (in the polynomial degree) stable in $L^2(\omega_i)$. The approximation ${\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v$ is now taken to be ${\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v := \sum_{i=0}^N \varphi_i J^i v$. The desired approximation properties follow now from [@babuska-melenk96 Theorem [2.1]{}].

The above proposition implies the following bounds.

\[cr:int\] For $v\in H^1_0(\Omega)$, the interpolant from Proposition \[pr:intp\] satisfies $${\left\|v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v\right\|}^2_{L^2(K_j)}\le
C^2_I\alpha_j{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_j}^2 ,
\qquad j=1,2,\ldots,N,$$ and $$\big|(v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v)(x_j)\big|^2\le
C_I^2\gamma_j\left({\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon},\widetilde K_{j}}^2+{\left|\!\left|\!\left|v\right|\!\right|\!\right|}^2_{{\epsilon},\widetilde K_{j+1}}\right),
\qquad j=1,2,\ldots,N-1.$$ Here, we let $$\begin{aligned}
\label{eq:alpha}
\alpha_j&=
\min\left\{{\epsilon}^{-1}h_j^2p_j^{-2},1\right\},\qquad j=1,\ldots, N,\end{aligned}$$ and, with $$\begin{aligned}
\beta_j&=\alpha_jh_j^{-1}+2\sqrt{{\epsilon}^{-1}\alpha_j},\label{eq:beta}\end{aligned}$$ we define $$\label{eq:gamma}
\gamma_j=\frac{\beta_j\beta_{j+1}}{\beta_j+\beta_{j+1}},$$ for $1\le j\le N-1$, and $\gamma_0=\gamma_N=0$. Moreover, $C_I$ is the constant from .

We proceed along the lines of [@Ve1998]. Using the bounds from Proposition \[pr:intp\], for each element $K_j\in{\mathcal{T}}$, we have that $$\begin{aligned}
{\left\|v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v\right\|}^2_{L^2(K_j)}&\le 
C^2_I\frac{h_j^2}{{\epsilon}p_j^2}{\epsilon}{\left\|v'\right\|}^2_{L^2(\widetilde K_j)}.\end{aligned}$$ Furthermore, $$\begin{aligned}
{\left\|v-{\pi_{{V^{\rm hp}({\mathcal{T}},{\bm p})}}}v\right\|}^2_{L^2(K_j)}
&\le C^2_I{\left\|v\right\|}_{L^2(\widetilde K_j)}^2.\end{aligned}$$ Combining these two estimates, yields the first bound.

In order to prove 