the form (\[XG3\]), with $\kappa=1$, therefore, by Proposition \[pro:1\] we have $$\label{KC5}
e_{2n}^{{\rm q }}(S_MR,B_{L_\infty^N})\le e_n^{{\rm q }}(S_M,B_{L_\infty^M}).$$ Moreover $$\begin{aligned}
S_MR(f)&=&\frac{1}{M}\sum_{j=0}^{M-1}f(\eta(j))=\frac{1}{M}\sum_{i=0}^{N-1}h(i)f(i)\\
&=&\frac{n}{M}\sum_{i=0}^{N-1}\tilde{g}(i)f(i)=\frac{nN}{M}S_{N,\tilde{g}}f.\end{aligned}$$ This together with (\[QC7\]) and (\[KC5\]) implies $$\begin{aligned}
e_{2n}^{{\rm q }}(S_{N,\tilde{g}},B_{L_\infty^N})&=&e_{2n}^{{\rm q }}\left(\frac{M}{nN}S_M R,B_{L_\infty^N}\right)
=\frac{M}{nN} e_{2n}^{{\rm q }}(S_M R,B_{L_\infty^N})\\
&\le& e_n^{{\rm q }}(S_M,B_{L_\infty^M})
\le cn^{-1},\end{aligned}$$ the latter relation being a consequence of [@BHM:00] (see also [@Hei01], Theorem 1, for the form stated here). Combining this with (\[QC6\]) and scaling the index gives the desired result.

Now we formally derive the complex case from the real case. Let $g\in L_1^N(\C)$ and let $g_1,g_2\in L_1^N({{\mathbb R}})$ be defined by $$\label{QV1}
g(j)=g_1(j)+\imath g_2(j)\quad(j=0,\dots,N-1,\;\imath=\sqrt{-1}).$$ Clearly, $$\label{QV2}
\|g_{\alpha }\|_{L_1^N({{\mathbb R}})}\le \|g\|_{ L_1^N(\C)}\quad ({\alpha }=1,2).$$ We shall express $\mathcal{P}=(B_{L_\infty^N},\C,S_{N,g},{{\mathbb K}},\Lambda)$ by the help of $$\mathcal{P}=(B_{L_\infty^N},{{\mathbb R}},S_{N,g_{{\alpha }}},{{\mathbb K}},\Lambda)\quad({\alpha }=1,2).$$ Define $R_1,R_2:L_\infty^N(\C)\to L_\infty^N({{\mathbb R}})$ for $f\in L_\infty^N(\C)$ by $$\label{QV6}
(R_1f)(j)={\rm Re}(f(j)),\quad (R_2f)(j)={\rm Im}(f(j))\quad(j=0,\dots,N-1).$$ Clearly, $R_1,R_2$ are of the form (\[XG3\]) and map $B_{L_\infty^N(\C)}$ to $B_{L_\infty^N({{\mathbb R}})}$. Define $J_{{\alpha }{\beta }}:{{\mathbb R}}\to \C$ (${\alpha },{\beta }\in \{1,2\}$) by $$J_{11}a=-J_{22}a=a,\quad J_{12}a=J_{21}a=\imath a \quad (a\in {{\mathbb R}}).$$ Then we have, by (\[QV1\]) and (\[QV6\]), $$S_{N,g}f=\sum_{{\alpha },{\beta }=1}^2 J_{{\alpha }{\beta }}S_{N,g_{\alpha }}R_{\beta }f.$$ Let $\nu$ be the smallest natural number with $e^{-\nu/8}\le 1/16$. By Proposition \[pro:3\] $$\label{QV4}
e_{2\nu n}(S_{N,g},B_{L_\infty^N(\C)})\le 