Native Matérn interpolation is an orthogonal projection with power function

The span $V_X$ has finite dimension and is therefore closed. Let ${\mathrm{\Pi }}_{X}f$ be the orthogonal projection of $f$ onto $V_X$. For each site $x_i$, the residual is orthogonal to $k(\cdot -x_i)$; by reproduction [F1], $(f-{\mathrm{\Pi }}_{X}f)(x_i)=0$. Thus ${\mathrm{\Pi }}_{X}f$ interpolates the data. Its coefficients solve $K_{X}c=(f(x_i){)}_i$, which has a unique solution by [F2], so it is the unique kernel interpolant.

Put $e=f-s_{X}f=(I-{\mathrm{\Pi }}_X)f$. Self-adjointness of $I-{\mathrm{\Pi }}_X$ and reproduction give $e(x)=⟨e,k(\cdot -x)⟩=⟨e,(I-{\mathrm{\Pi }}_X)k(\cdot -x)⟩\mathrm{.}$ Cauchy--Schwarz [F4] yields $\mathrm{\mid }e(x)\mathrm{\mid }\le \mathrm{\parallel }e\mathrm{\parallel }{P}_X(x)\le \mathrm{\parallel }f\mathrm{\parallel }{P}_X(x)$ by contractivity of the orthogonal projection.

The projection of $k(\cdot -x)$ has the form ${\sum }_j{v}_{j}k(\cdot -x_j)$. Reproduction and orthogonality at all sites give $K_{X}v=k_X(x)$, so $v=K_X^{-1}{k}_X(x)$ by [F2]. Expanding the squared residual norm and using this equation gives $P_X(x{)}^2=\mathrm{\parallel }k(\cdot -x){\mathrm{\parallel }}^2-\mathrm{\parallel }{\mathrm{\Pi }}_{X}k(\cdot -x){\mathrm{\parallel }}^2=k(0)-k_X(x{)}^{\ast }{K}_X^{-1}{k}_X(x)\mathrm{.}$ It is nonnegative because it is a squared norm.