and its analogous for $\xi (x)$ (see the proof of Theorem \[th1\], formula \[xi-sim\]) we obtain

$$\begin{aligned}
\Vert T_{N,M}(x)-\xi (x) \Vert ^{2} _{\alpha , 2}
 &=&\sum_{m\geq 0} (3+m) ^{\alpha } \frac{m!} {(m+1) (m+2)} \sum_{k,l=1; k\not= l}^{m+2}
\int_{[0,1] ^{2m}} dx_{1}..dx_{m+2} dy_{1}..dy_{m+2} \\
&& \left(  \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}a_{m}^{x}(s_{i}, t_{j}) 1_{[0,s_{i}]} ^{\otimes m+1} (x_{1},..,\overline{x_{k}},.., x_{m+2})1_{[s_{i},s_{i+1}]}(x_{k}) 1_{[0,t_{j}]} ^{\otimes m+1} (y_{1},..,\overline{y_{l}} ,..y_{m+2}) \right. \\
&& \left. -  a_{m}^{x} (x_{k}, y_{l}) 1_{[0,x_{k} ]}^{\otimes m+1} (x_{1},..,\overline{x_{k}},.., x_{m+2})1_{[0,y_{l}]}^{\otimes m+1} (y_{1},..,\overline{y_{l}} ,..y_{m+2}) \right)^{2}\end{aligned}$$

and as in the proof of Theorem \[th1\] $$\begin{aligned}
\Vert T_{N,M}(x)-\xi (x) \Vert ^{2} _{\alpha , 2}
&=&\sum_{m\geq 0} (3+m) ^{\alpha }\frac{m!} {(m+1) (m+2)} \sum_{k,l=1; k\not= l}^{m+2} \\
 &&\left( \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}(a_{m}^{x}(s_{i}, t_{j}))^{2} s_{i}^{m+1} (s_{i+1}-s_{i}) t_{j}^{m+1} (t_{j+1}-t_{j})  \right. \\
&&\left. - 2 \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}a_{m}^{x}(s_{i}, t_{j})s_{i}^{m+1}  t_{j}^{m+1}\int_{s_{i}}^{s_{i+1}} \int_{t_{j}}^{t_{j+1}}dx_{k}dy_{l}a_{m}^{x} (x_{k}, y_{l})  \right. \\
&&\left. + \int_{0}^{1} \int_{0}^{1} dx_{k}dy_{l} (a_{m}^{x}(x_{k}, y_{l})) ^{2} x_{k}^{m+1} y_{l} ^{m+1} \right)\\
&=&\sum_{m\geq 0} (3+m) ^{\alpha }m! \left( \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}(a_{m}^{x}(s_{i}, t_{j}))^{2} s_{i}^{m+1} (s_{i+1}-s_{i}) t_{j}^{m+1} (t_{j+1}-t_{j})  \right. \\
&&\left. -  2 \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}a_{m}^{x}(s_{i}, t_{j})s_{i}^{m+1}  t_{j}^{m+1}\int_{s_{i}}^{s_{i+1}} \int_{t_{j}}^{t_{j+1}}dudv a_{m}^{x} (u,v)\right. \\
&&\left.  +  \int_{0}^{1} \int_{0}^{1}dudv (a_{m}^{x}(u,v))^{2} (uv) ^{m+1}\right).\end{aligned}$$ Now we claim that, for every fixed $m\geq 0$, the sequence $$\begin{aligned}
&&\left( \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}(a_{m}^{x}(s_{i}, t_{j}))^{2} s_{i}^{m+1} (s_{i+1}-s_{i}) t_{j}^{m+1} (t_{j+1}-t_{j})  \right. \\
&&\left. -  2 \sum_{i=0}^{N-1} \sum_{j=0}^{M-1}a_{m}^{x}(s_{i}, t_{j})s_{i}^{m+1}  t_{j}^{m+1}\int_{s_{i}}^{s_{i+1}} \int_{t_{j}}^{t_{j+1}}dudv a_{m}^{x} (u,v) 