Tiniest tensor

\(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\max_i(p_i)\leqslant k\right\}\)

\(\mathcal{Z}^{(30)}_k=\left\{x\tilde{B}_{k+1}(y),(1-x)\tilde{B}_{k+1}(y),y\tilde{B}_{k+1}(x),(1-y)\tilde{B}_{k+1}(x)\right\} \)
where

• \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\)
• \(\tilde{B}_k\) is \((\tilde{P}_{k+2}-\tilde{P}_k)/(4k+6)\)

\(\mathcal{Z}^{(31)}_k=\left\{xy\tilde{B}_{k+1}(z),(1-x)y\tilde{B}_{k+1}(z),(x(1-y)\tilde{B}_{k+1}(z),(1-x)(1-y)\tilde{B}_{k+1}(z)\right\} \)
where

• \(\tilde{P}_k\) is the degree \(k\) Legendre polynomial on \([0,1]\)
• \(\tilde{B}_k\) is \((\tilde{P}_{k+2}-\tilde{P}_k)/(4k+6)\)

\(\mathcal{Z}^{(32)}_k=\left\{xz\tilde{B}_{k+1}(y),(1-x)z\tilde{B}_{k+1}(y),(x(1-z)\tilde{B}_{k+1}(y),(1-x)(1-z)\tilde{B}_{k+1}(y)\right\}\)

\(\mathcal{Z}^{(33)}_k=\left\{yz\tilde{B}_{k+1}(x),(1-y)z\tilde{B}_{k+1}(x),(y(1-z)\tilde{B}_{k+1}(x),(1-y)(1-z)\tilde{B}_{k+1}(x)\right\}\)