Interpolatory Rules for $U_n$

Let $T_{n-1}$ be the $n-1$-simplex defined by $T_{n-1}=\{{\bf x} \vert {\bf x}\in {\cal R}^{n-1},0\leq x_1+x_2\cdots+x_{n-1}\leq 1\},$ and, for any ${\bf x}\in T_{n-1}$, define $x_n = 1 - x_1-x_2\cdots-x_{n-1}$. Also, let ${\bf t}_{\bf p}=(t_{p_1}, t_{p_2}, \ldots, t_{p_n})$, and let ${\bf p}=(p_1, p_2, \ldots, p_n)$. If real numbers $t_0, t_1, \ldots, t_m$ are given, satisfying the condition

$$\vert{\bf t}_{{\bf p}}\vert = t_{p_1}+t_{p_2}+\cdots+t_{p_n} ... ...\mbox{whenever} \vert{\bf p}\vert = p_1+p_2+\cdots+p_n = m,$$

for non-negative integers $p_1, \ldots, p_n$, then the Lagrange interpolation formula (Sylvester [8]) for a function $g({\bf x})$ on $T_{n-1}$ is given by

$$L^{(m,n-1)}(g,{\bf x}) = \sum_{\vert{\bf p}\vert=m}\prod_{i=... ...j=0}^{p_i-1}\frac{x_i-t_j}{t_{p_i}-t_j} g({\bf t}_{\bf p}).$$

$L^{(m,n-1)}(g,{\bf x})$ is the unique polynomial of degree $m$ which interpolates $g({\bf x})$ at all of the $m+n-1\choose m$ points in the set $\{{\bf x} \vert {\bf x}=(t_{p_1}, \ldots, t_{p_{n-1}}), \vert{\bf p}\vert = m\}$. Sylvester provided families of points, satisfying the condition $\vert{\bf t}_{\bf p}\vert=1$ when $\vert{\bf p}\vert=m$, in the form $t_i = \frac{i+\mu}{m+\mu n}$, for $i=0,1,\ldots,m$, and $\mu$ real. If $0 \leq \mu \leq 1$, all interpolation points for $L^{(m,n-1)}(g,{\bf x})$ are in $T_{n-1}$. Sylvester derived families of interpolatory rules for integration over $T_{n-1}$ by integrating $L^{(m,n-1)}(g,{\bf x})$.

Fully symmetric interpolatory integration rules for $U_n$ can be obtained by a simple change of variables. Make the substitutions $x_i = z_{i}^2$, and $t_i = u_{i}^2$ in $L^{(m,n-1)}(g,{\bf x})$, and define

$$M^{(m,n)}(f,{\bf z}) = \sum_{\vert{\bf p}\vert=m} \prod_{i=1... ...}\frac{z_i^2-u_j^2}{u_{p_i}^2-u_j^2} f\{{\bf u}_{\bf p}\}.$$

where $f\{{\bf u}\}$ is a symmetric sum defined by

$$f\{{\bf u}\} = 2^{-c({\bf u})} \sum_{\bf s}f( s_1 u_1, s_2 u_2,\ldots, s_n u_n ),$$

with $c({\bf u})$ the number of nonzero entries in $(u_1,u_2,\ldots,u_n)$, and the sum $\sum_{\bf s}$ taken over all of the sign combinations that occur when $s_i = \pm 1$, for those $i$ with $u_i \neq 0$.

Theorem 2.1   If

$$w_{\bf p}= I(\prod_{i=1}^n \prod_{j=0}^{p_i-1}\frac{z_i^2-u_j^2}{u_{p_i}^2-u_j^2}),$$

then

$$Q^{(m,n)}(f) = \sum_{\vert{\bf p}\vert=m}w_{{\bf p}}f\{{\bf u}_{\bf p}\}.$$

is an integration rule of polynomial degree $2m+1$ for $U_n$.

Proof. Let ${\bf z}^{{\bf k}}=z_1^{k_1}z_2^{k_2}\cdots z_n^{k_n}$. $I$ and $R$ are both linear functionals, so it is sufficient to show that $Q^{(m,n)}({\bf z}^{{\bf k}})=I({\bf z}^{{\bf k}})$ whenever $\vert{\bf k}\vert \leq 2m+1$. If ${\bf k}$ has any component $k_i$ that is odd, then $I({\bf z}^{{\bf k}}) = 0$, and $Q^{(m,n)}({\bf z}^{{\bf k}}) = 0$ because every term ${\bf u}_{{\bf q}}^{{\bf k}}$ in each of the symmetric sums $f\{{\bf u}_{\bf p}\}$ has a canceling term $-{\bf u}_{{\bf q}}^{{\bf k}}$. Therefore, the only monomials that need to be considered are of the form ${\bf z}^{2{\bf k}}$, with $\vert{\bf k}\vert \leq m$. The uniqueness of $L^{(m,n-1)}(g,{\bf x})$ implies $L^{(m,n-1)}({\bf x}^{{\bf k}},{\bf x})={\bf x}^{{\bf k}}$ whenever $\vert{\bf k}\vert \leq m$, so $M^{(m,n)}({\bf z}^{2{\bf k}},{\bf z}) = {\bf z}^{2{\bf k}}$, whenever $\vert{\bf k}\vert \leq m$. Combining these results:

$$\begin{eqnarray*} I(f) &=& I(M^{(m,n)}(f,{\bf z}) ) \\ &=& \sum_{\vert{\bf p}\v... ...bf p}\vert=m}w_{{\bf p}}f\{{\bf u}_{\bf p}\}\\ &=& Q^{(m,n}(f), \end{eqnarray*}$$

whenever $f({\bf z}) = {\bf z}^{{\bf k}}$, with $\vert{\bf k}\vert \leq 2m+1$, so $Q^{(m,n)}(f)$ has polynomial degree $2m+1$.