Matérn interpolation and its power function converge at rate h^ν

By [F4] choose an integer $m>\nu$ and apply the zeros estimate [F2] with this fixed $m$. Regard $f$ as an element of the same set of continuous restrictions $H^s(\mathrm{\Omega })={\mathcal{H}}_k(\mathrm{\Omega })$ using [F1], and let $e=f-s_{X}f$. By [F3], $e{\mathrm{\mid }}_X=0$ and $\mathrm{\parallel }e{\mathrm{\parallel }}_{{\mathcal{H}}_k(\mathrm{\Omega })}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathcal{H}}_k(\mathrm{\Omega })}$. Therefore [F1] and [F2] give $\mathrm{\parallel }e{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C_0{h}_X^{\nu }\mathrm{\parallel }e{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}\le C_0{c}^{-1}{h}_X^{\nu }\mathrm{\parallel }e{\mathrm{\parallel }}_{{\mathcal{H}}_k(\mathrm{\Omega })}\le C_0{c}^{-1}{C}_1{h}_X^{\nu }\mathrm{\parallel }f{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}\mathrm{.}$ This is the first estimate.

Fix $x\in \mathrm{\Omega }$ and set $r_x=(I-{\mathrm{\Pi }}_X)k(\cdot -x)$, so $P_X(x)=\mathrm{\parallel }{r}_x\mathrm{\parallel }$. Self-adjointness and reproduction in [F3] give $f(x)-s_{X}f(x)=⟨f,r_x⟩$. The bound in [F3] shows that this residual evaluation functional has norm at most $P_X(x)$. If $r_x=0$, it is the zero functional and its norm is $P_X(x)=0$. If $r_x\mathrm{\ne }0$, the unit vector $f_x=r_x\mathrm{/}\mathrm{\parallel }{r}_x\mathrm{\parallel }$ satisfies $f_x(x)-s_X{f}_x(x)=⟨f_x,r_x⟩=\mathrm{\parallel }{r}_x\mathrm{\parallel }=P_X(x)$, so the functional norm is exactly $P_X(x)$. Applying step 1.1 to every $f$ of native norm at most one therefore gives $P_X(x)\le C_2{h}_X^{\nu }$, uniformly in $x$; taking the supremum proves the second estimate. The constant does not depend on $X$ except through the upper bound on $h_X$, and no mesh ratio entered either [F2] or this argument.