Key idea:
V = V_x\otimes V_y\otimes V_z
\begin{align*} a^{(q)} &= (a^{(q)}_0,\ldots,a^{(q)}_{N_q}),\quad q=0,\ldots,m\\ p &= a^{(0)}\otimes\cdots\otimes a^{(m)}\\ p_{i_0,i_1,\ldots,i_m} &= a^{(0)}_{i_1}a^{(1)}_{i_1}\cdots a^{(m)}_{i_m} \end{align*}
\begin{align*} \baspsi_{p,q,r}(x,y,z) &= \hat\baspsi_p(x)\hat\baspsi_q(y)\hat\baspsi_r(z)\\ u(x,y,z) &= \sum_{p\in\Ix}\sum_{q\in\Iy}\sum_{r\in\Iz} c_{p,q,r} \baspsi_{p,q,r}(x,y,z) \end{align*}