The normalized Matérn kernel in the 2π Fourier convention

Fix the dimension $d\ge 1$ of an admitted domain, a smoothness parameter $\nu >0$, and an inverse length $a>0$. Assume Countable Choice for the Lebesgue-integration suppliers used to justify this formula; this definition does not assume full Choice. Put $s=\nu +d\mathrm{/}2$ and $A=\frac{\mathrm{\Gamma }(s)(4\pi {)}^{d\mathrm{/}2}{a}^{2\nu }}{\mathrm{\Gamma }(\nu )},w(\xi )=A(a^2+4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{-s}\mathrm{.}$ All integrals are Lebesgue integrals with respect to ${\lambda }_d$ (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$), and $\mathrm{\Gamma }$ is the Euler integral of The real Gamma function by Euler's integral. With the negative-sign Fourier transform convention $\widehat{U}(\xi )=\int U(x)e^{-2\pi ix\cdot \xi }dx$, define the normalized stationary kernel by $k(z)={\int }_{{\mathbb{R}}^d}w(\xi )e^{2\pi iz\cdot \xi }d\xi ,K(x,y)=k(x-y)\mathrm{.}$ The corresponding length parameter in the usual Matérn notation is $\mathrm{\ell }=\sqrt{2\nu }\mathrm{/}a$. For $b>0$, define the modified Bessel function in its integral normalization by $K_{\nu }(b)=\frac{1}{2}{\left(\frac{b}{2}\right)}^{\nu }{\int }_0^{\mathrm{\infty }}{u}^{-\nu -1}{e}^{-u-b^2\mathrm{/}(4u)}du\mathrm{.}$ The classical Bessel expression for $k(z)$ when $z\mathrm{\ne }0$, its value $k(0)=1$, absolute convergence, continuity, and radial symmetry are established by The Matérn spectral integral, normalization, and radial formula ↗. This definition makes no finite-precision stability claim.