Gaussian RKHS power function decays exponentially with inverse fill distance

Let $z_c$ be the center of $\mathrm{\Omega }$, let $L>0$ be its side length, and use orthonormal coordinates along its edges, writing $u(z)$ for the coordinates of $z-z_c$. These coordinates preserve distances and give $\mathrm{\parallel }u(z){\mathrm{\parallel }}_2\le B=L\sqrt{d}\mathrm{/}2$ for every $z\in \mathrm{\Omega }$. Put $t=B^2\mathrm{/}{\sigma }^2>0$. For a multi-index $\alpha =({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{N}}_0^d$, write $\mathrm{\mid }\alpha \mathrm{\mid }={\sum }_i{\alpha }_i$, $\alpha !={\prod }_i{\alpha }_i!$, and $u^{\alpha }={\prod }_i{u}_i^{{\alpha }_i}$, and define ${\mathrm{\Phi }}_{\alpha }(z)=\exp (-\mathrm{\parallel }u(z){\mathrm{\parallel }}_2^2\mathrm{/}(2{\sigma }^2))\frac{u(z{)}^{\alpha }}{{\sigma }^{\mathrm{\mid }\alpha \mathrm{\mid }}\sqrt{\alpha !}}\mathrm{.}$ Multiplying the convergent exponential power series in the $d$ coordinates yields ${\sum }_{\alpha }{\mathrm{\Phi }}_{\alpha }(z){\mathrm{\Phi }}_{\alpha }(w)=e^{-(\mathrm{\parallel }u(z){\mathrm{\parallel }}_2^2+\mathrm{\parallel }u(w){\mathrm{\parallel }}_2^2)\mathrm{/}(2{\sigma }^2)}{e}^{u(z)\cdot u(w)\mathrm{/}{\sigma }^2}=k_{\sigma }(z,w)\mathrm{.}$ The series is absolutely convergent, since replacing each coordinate product by its absolute value gives the finite value $e^{-(\mathrm{\parallel }u(z){\mathrm{\parallel }}_2^2+\mathrm{\parallel }u(w){\mathrm{\parallel }}_2^2)\mathrm{/}(2{\sigma }^2)}\exp ({\sum }_i\mathrm{\mid }{u}_i(z)u_i(w)\mathrm{\mid }\mathrm{/}{\sigma }^2)$. In particular the vector $\mathrm{\Phi }(z)=({\mathrm{\Phi }}_{\alpha }(z){)}_{\alpha }$ is square summable with squared norm $k_{\sigma }(z,z)=1$.

For any finite set of points $z_1,\dots ,z_q\in \mathrm{\Omega }$ and real coefficients $a_1,\dots ,a_q$, the reproducing property gives ${\parallel {\sum }_{j=1}^q{a}_j{k}_{\sigma }(\cdot ,z_j)\parallel }_{{\mathcal{H}}_{\sigma }}^2={\sum }_{j,l=1}^q{a}_j{a}_l{k}_{\sigma }(z_j,z_l)={\parallel {\sum }_{j=1}^q{a}_j\mathrm{\Phi }(z_j)\parallel }_{{\mathrm{\ell }}^2}^2\mathrm{.}$ The second equality follows from step 1.1 by interchanging a finite sum with an absolutely convergent series. Thus this norm identity concerns finite kernel combinations and requires no description of the whole Gaussian RKHS or extension of its functions to ${\mathbb{R}}^d$. The RKHS used here is available under the stated Countable Choice hypothesis.

Fix $X$ as in the Statement, set $h=h_X$, and for now choose an integer $n\ge 1$ such that $2nh\mathrm{/}L\le \kappa$. Fix any $x\in \mathrm{\Omega }$ and take coefficients ${\lambda }_j$ from [F4]. Define real coefficients $b_j={\lambda }_j\exp \left(\frac{\mathrm{\parallel }u(x_j){\mathrm{\parallel }}_2^2-\mathrm{\parallel }u(x){\mathrm{\parallel }}_2^2}{2{\sigma }^2}\right)\mathrm{.}$ Then ${\sum }_j\mathrm{\mid }{b}_j\mathrm{\mid }\le e^{t\mathrm{/}2}{A}^n$. For every $\mathrm{\mid }\alpha \mathrm{\mid }\le n$, the polynomial $z\mapsto u(z{)}^{\alpha }$ has total degree at most $n$, so reproduction implies ${\sum }_j{b}_j{\mathrm{\Phi }}_{\alpha }(x_j)=\frac{e^{-\mathrm{\parallel }u(x){\mathrm{\parallel }}_2^2\mathrm{/}(2{\sigma }^2)}}{{\sigma }^{\mathrm{\mid }\alpha \mathrm{\mid }}\sqrt{\alpha !}}{\sum }_j{\lambda }_{j}u(x_j{)}^{\alpha }={\mathrm{\Phi }}_{\alpha }(x)\mathrm{.}$ Consequently all coordinates of $\mathrm{\Phi }(x)-{\sum }_j{b}_j\mathrm{\Phi }(x_j)$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le n$ vanish.

Write ${\mathrm{\Phi }}_{>n}(z)$ for the coordinates with $\mathrm{\mid }\alpha \mathrm{\mid }>n$. Expanding $({\sum }_i{u}_i(z{)}^2\mathrm{/}{\sigma }^2{)}^r$ by multiplication shows that its monomial coefficient at multi-index $\alpha$ of size $r$ is $r!\mathrm{/}\alpha !$: it counts the ordered $r$ choices of coordinates with the prescribed multiplicities. Hence $\mathrm{\parallel }{\mathrm{\Phi }}_{>n}(z){\mathrm{\parallel }}_{{\mathrm{\ell }}^2}^2=e^{-\mathrm{\parallel }u(z){\mathrm{\parallel }}_2^2\mathrm{/}{\sigma }^2}{\sum }_{r=n+1}^{\mathrm{\infty }}\frac{(\mathrm{\parallel }u(z){\mathrm{\parallel }}_2^2\mathrm{/}{\sigma }^2{)}^r}{r!}\le {\sum }_{r=n+1}^{\mathrm{\infty }}\frac{t^r}{r!}\mathrm{.}$ If $n+2\ge 2t$, the ratio of any successive terms in the last tail is at most $1\mathrm{/}2$, so summing the geometric majorant gives ${\sum }_{r=n+1}^{\mathrm{\infty }}\frac{t^r}{r!}\le \frac{2t^{n+1}}{(n+1)!}\le 2{\left(\frac{et}{n+1}\right)}^{n+1}\mathrm{.}$ For the final inequality, $\log (q!)={\sum }_{j=1}^q\log j\ge {\int }_1^q\log sds=q\log q-q+1$ for every integer $q\ge 1$, and exponentiation yields $q!\ge (q\mathrm{/}e{)}^q$. Thus the bound holds uniformly for every $z\in \mathrm{\Omega }$.

By [F3], the infimum defining $P_X(x)$ is attained by orthogonal projection onto the finite section span. The vector ${\sum }_j{b}_j{k}_{\sigma }(\cdot ,x_j)$ is in that span, so its error norm is an upper bound for $P_X(x)$. Steps 2.1 and 2.2 show that its error norm is $\mathrm{\parallel }{\mathrm{\Phi }}_{>n}(x)-{\sum }_j{b}_j{\mathrm{\Phi }}_{>n}(x_j){\mathrm{\parallel }}_{{\mathrm{\ell }}^2}$. Apply the triangle inequality to these finitely many square-summable vectors and use steps 2.2 and 2.3 to obtain $P_X(x)\le \sqrt{2}(1+e^{t\mathrm{/}2}{A}^n){\left(\frac{et}{n+1}\right)}^{(n+1)\mathrm{/}2}\mathrm{.}$ The ${\mathrm{\ell }}^2$ triangle inequality here can be obtained from the finite Euclidean triangle inequality on every initial finite collection of multi-indices and then passing to the supremum of their squared partial sums; thus no additional choice principle is involved. The bound is independent of $x$, and holds whenever $n\ge 1$, $2nh\mathrm{/}L\le \kappa$, and $n+2\ge 2t$.

Put $D=\kappa L\mathrm{/}2>0$ and $n=\lfloor D\mathrm{/}h\rfloor$. For all sufficiently small $h>0$ this integer satisfies $n\ge 1$, $n+2\ge 2t$, and $D\mathrm{/}(2h)\le n\le D\mathrm{/}h$; the lower bound follows from $\lfloor s\rfloor \ge s\mathrm{/}2$ for $s\ge 2$. Also $2nh\mathrm{/}L\le \kappa$. Since $1+e^{t\mathrm{/}2}{A}^n\le 2e^{t\mathrm{/}2}{A}^n$, the logarithm of the positive bound in step 3.1 is at most $-\frac{1}{2}(n+1)\log (n+1)+K(n+1),$ where one may take $K=1+\frac{3}{2}\log 2+t\mathrm{/}2+\log A+\frac{1}{2}\mathrm{\mid }\log (et)\mathrm{\mid }>0$. Decrease $h_0>0$ until for all $0<h\le h_0$ the preceding integer conditions hold, $h<1$, $\log (n+1)\ge 4K$, and $\log (n+1)\ge \frac{1}{2}\mathrm{\mid }\log h\mathrm{\mid }$. Such an $h_0$ exists because $n\ge D\mathrm{/}(2h)$ implies $\log (n+1)\ge \mathrm{\mid }\log h\mathrm{\mid }+\log (D\mathrm{/}2)$, which tends to infinity and eventually exceeds $\mathrm{\mid }\log h\mathrm{\mid }\mathrm{/}2$.

For $0<h\le h_0$ the exponent in step 4.1 is at most $-(n+1)\log (n+1)\mathrm{/}4$, and hence at most $-D\mathrm{\mid }\log h\mathrm{\mid }\mathrm{/}(16h)$, by the two lower bounds for $n$ and $\log (n+1)$. Therefore step 3.1 gives, uniformly in $x\in \mathrm{\Omega }$, $P_X(x)\le \exp (-\frac{D}{16}\frac{\mathrm{\mid }\log h\mathrm{\mid }}{h})\mathrm{.}$ Taking the supremum over $x$ proves the Statement with $c=D\mathrm{/}16$ and $C=1$, and also with any larger positive $C$. All constants depend only on $d$, $\mathrm{\Omega }$, and $\sigma$. Countable Choice was used only through the canonical RKHS supplier in step 2.1; the reproduction and feature estimates add no further choice assumption.