Let $\Omega = [0, 1]^d$ and suppose that $f \colon \Omega \to \mathbb{R}$ lies in order $m > d/2$ Sobolev space; i.e., $$ \|f\|_{H^m(\Omega)}^2 = \sum_{|\alpha| \leq m} \|D^\alpha f\|_{L^2(\Omega)}^2 < \infty. $$ Let $x_1, \dots, x_n\in \Omega$ and define the distance $$ h \equiv h_\Omega(x_1, \dots, x_n) = \sup_{y \in \Omega} \min_{i = 1, \dots, n} |y - x_i|, $$ where above $|\cdot|$ denotes the usual Euclidean norm on $[0, 1]^d$.
We can then consider the minimum Sobolev norm interpolant: $$ s_{m, \Omega}(x_1, \dots, x_n) = \mathrm{argmin}_{g \in L^2(\Omega)} \Big\{\|g\|_{H^m(\Omega)} : g(x_i) = f(x_i), ~~\mbox{for}~i = 1,\dots n\Big\} $$
I have seen the following theorem quoted in various papers/sources (Matveev, the book by Anatoly Yu. Bezhaev, Vladimir A. Vasilenko, etc.), but none containing a full proof:
Theorem: Suppose that $f \in H^m(\Omega)$ where $\Omega = [0, 1]^d$ and $m > d/2$. Let $x_1, \dots, x_n \in \Omega$ and let $\widehat{f}$ denote the interpolant $s_{m, \Omega}(x_1, \dots, x_n)$. Then there exists a constant $C$ depending only on $\Omega, m$ such that $$ \|f - \widehat{f}\|_{L^2(\Omega)} \leq C~ h^{m}~\|f\|_{H^m(\Omega)}. $$
Is there any source (in English) which provides a full (ideally self-contained) proof of this result?
