An example of the k-th order spline representation: k=2, d=3

Consider the case that \(d=3\) and \(k=2\). Then, \[ \begin{eqnarray*} Q(x)&=&\sum_{s_1\subset\{1,2,3\},\mid s_1\mid>0} \phi^1_0(x(s_1)) Q^{(1)}_{s_1}(0(s_1))\\ &&+\sum_{\bar{s}_2,\mid s_2\mid>0}\phi^1_0(x(s_1/s_2))\phi^2_0(x(s_2))Q^{(2)}_{\bar{s}(2)}(0(s_2))\\ &&+\sum_{\bar{s}(3), \mid s_3\mid>0}\phi^1_0(x(s_1/s_2))\phi_0^2(x(s_2/s_3))\int \phi^2_{u(s_3)}(x(s_3)) Q^{(2)}_{\bar{s}(3)}(du(s_3)). \end{eqnarray*} \] Note that in this example, the parametric component of the model consists of two parts. The first part is the fully saturated interaction model, which includes the terms \[ x_1,x_2,x_3,x_1x_2,x_1x_3,x_2x_3,x_1x_2x_3. \] The second component is a parametric model of the form \[ \sum_{s_1,s_2\subset s_1,\mid s_2\mid>0}\prod_{j\in s_1/s_2}x_j \prod_{j\in s_2}x_j^2. \] Let \((x-u)^2_+=I(x\geq u)(x-u)^2\). Now, we enumerate all terms from this second parametric part:

The infinite dimensional component includes a union over \(s_3,\mid s_3\mid>0\) and \(u(s_3)\) of a \((s_3,u(s_3))\)-specific saturated interaction model given by \[ \sum_{\{ (s_1,s_2): s_2\subset s_1,s_3\subset s_2}\prod_{j\in s_1/s_2}x_j\prod_{j\in s_2/s_3}x_j^2\prod_{j\in s_3}I(x_j>u)(x_j-u)^2. \] We now enumerate all terms from this infinite dimensional component: