Series least squares with a spline basis

Let $n,K\ge 1$, let $p_K(x)=(s_1(x),\dots ,s_K(x){)}^T$ be a specified $K$-dimensional spline basis on the data domain, and let $(X_i,Y_i{)}_{i=1}^n$ be observed real-valued data with inputs in that domain. In particular, one may specify the degree-$p$ B-spline basis $p_K(x)=(B_{1,p,\xi }(x),\dots ,B_{K,p,\xi }(x){)}^T$. Form the design matrix $P\in {\mathbb{R}}^{n\times K}$ with row $p_K(X_i{)}^T$. The series least-squares fitted training vector is ${\widehat{\mathbf{g}}}_K={\mathrm{\Pi }}_{\mathrm{col}(P)}\mathbf{Y},$ the orthogonal projection of $\mathbf{Y}=(Y_1,\dots ,Y_n{)}^T$ onto the column space of $P$. Conditional on the realized design, this is the fixed-design least-squares fitted vector of Fixed-design least squares as orthogonal projection. Equivalently, it is $P\widehat{\beta }$ for any minimizer $\widehat{\beta }$ of $\mathrm{\parallel }\mathbf{Y}-P\beta {\mathrm{\parallel }}_2^2$; all such minimizers have the same fitted training vector. The training prediction loss relative to a specified target vector ${\mathbf{g}}_0=(g_0(X_1),\dots ,g_0(X_n){)}^T$ is $n^{-1}\mathrm{\parallel }{\widehat{\mathbf{g}}}_K-{\mathbf{g}}_0{\mathrm{\parallel }}_2^2$. This definition specifies training fitted values and makes no test-risk assertion.