 that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{0\pi}\, h)^{q_k} \,n_k \beta_k\bigg| {\leqslant}C \left(  n_{k} h^{q_{k}+0} + \sqrt{n_kh^{q_k}\log n_k} \right),\end{aligned}$$ which completes the proof.  Proof of 