Expand rationomial sets on pyramids to include derivatives

[mscroggs]

As highlighted in #935, the current sets of rationomials on pyramids do not include some of the derivatives. For example, the function $xy/(1-z)$ is in a degree 2 space on a pyramid, but the $x$-derivative of this (ie $y/(1-z)$) is not in the rationomial space of any degree.

I propose expanding the rationomial spaces from their current definition
$$\text{span}\left\{\frac{x^{p_0}y^{p_1}z^{p_2}}{(1-z)^{\min(p_0,p_1)}}\,\middle|\,p_0,p_1,p_2\in\mathbb{N}_0,\,p_0+p_2\leqslant k,\,p_1+p_2\leqslant k\right\}$$
to the larger space
$$\text{span}\left\{\frac{x^{p_0}y^{p_1}z^{p_2}}{(1-z)^{p_3}}\,\middle|\,p_0,p_1,p_2,p_3\in\mathbb{N}_0,\,p_0+p_2\leqslant k,\,p_1+p_2\leqslant k,\,p_2+p_3\leqslant k\right\}.$$
For these larger spaces include, all derivatives of functions in the degree $k$ space are included in the degree $k+1$ space.

[mscroggs]

I took a closer look at this and realised that
$$\text{span}\left\{\frac{x^{p_0}y^{p_1}z^{p_2}}{(1-z)^{p_3}}\,\middle|\,p_0,p_1,p_2,p_3\in\mathbb{N}_0,\,p_0+p_2\leqslant k,\,p_1+p_2\leqslant k,\,p_2+p_3\leqslant k\right\}\not\subset L^2,$$
so propose the alternative space
$$\text{span}\left\{\frac{x^{p_0}y^{p_1}z^{p_2}}{(1-z)^{p_3}}\,\middle|\,p_0,p_1,p_2,p_3\in\mathbb{N}_0,\,p_0+p_2\leqslant k,\,p_1+p_2\leqslant k,\,p_2+p_3\leqslant k\right\}\cap L^2.$$

[mscroggs]

The previous proposal includes some higher degree rationomials in a degree k space. Better proposal is included in https://doi.org/10.5281/zenodo.15641641