Consider a function $f:{\mathbb{R}}^{\mathbb{2}}\to \mathbb{C}$. We usually represent such a function as $f(x,y)=u(x,y)+iv(x,y)$ for $u,v:{\mathbb{R}}^{\mathbb{2}}\to \mathbb{R}$ . Now, we could alternatively have chosen our basis of $\mathbb{C}$ to be $W=\{[1,1],[1,-1]\}$. Now we let $[v{]}_W={\alpha }_{1}z+{\alpha }_2\overline{z}$, in which case we are now choosing to represent $f([v{]}_W)=u({\alpha }_1,{\alpha }_2)+iv({\alpha }_1,{\alpha }_2)$ for new functions $u,v$.