A mean zero spatial process $X\left(\mathbf{t}\right)$ on the $n$-dimensional Euclidean space ${\mathbb{R}}^n$ is isotropic if its covariance function (c.f.) is of the form: $R_n\left(\tau \right)=\mathrm{Cov}[X\left(\mathbf{t}\right),X\left(\mathbf{s}\right)]$, where $\tau =|\mathbf{t}-\mathbf{s}|$, $\mathbf{t}$, $\mathbf{s}$ $\in {\mathbb{R}}^n$, and $R_n$ is an admissible function on ${\mathbb{R}}^1$. An isotropic spatial process $X$ has a bounded range of dependence if $sup\{\tau :R_n\left(\tau \right)\ne 0\}$ $<\infty$. Here we consider a class of isotropic c.f.’s $R_n\left(\tau \right)$, $n=1,2,\dots$ with bounded ranges of dependence, among which there are $R_1$, the classical triangular c.f. on the real line ($n=1$), and $R_3$, the spherical c.f. in dimension three ($n=3$). For each dimension $n\ge 1$, the admissibility of $R_n\left(\tau \right)$ as a c.f. in higher dimensions is studied. While it is well known that for each $n\ge 1$, $R_n$ is a legitimate c.f. on ${\mathbb{R}}^m$ for all $m\le n$ but it is shown that the considered $R_n$ is not a legitimate c.f. on ${\mathbb{R}}^m$ when $m>n$. Thus the spherical c.f. $R_3$ cannot be a c.f. on ${\mathbb{R}}^n$ when $n>3$. The issue of recognition of an isotropic c.f. on ${\mathbb{R}}^n$ is discussed, and simple procedures of constructing isotropic c.f.’s on ${\mathbb{R}}^n$ for every $n\ge 1$ are given. This article serves as one more reminder that caution must be taken concerning the legitimacy of a selected c.f. in the corresponding spatial dimensions.

平均零空间过程 $X\left(\mathbf{吨}\right)$ 在 $n$维欧几里得空间 ${\mathbb{电阻}}^n$ 如果其协方差函数 (cf) 具有以下形式，则为各向同性： ${\mathrm{电阻}}_n\left(\tau \right)=\mathrm{冠状病毒}[X\left(\mathbf{吨}\right),X\left(\mathbf{秒}\right)]$， 在哪里 $\tau =|\mathbf{吨}-\mathbf{秒}|$, $\mathbf{吨}$, $\mathbf{秒}$ $\in {\mathbb{电阻}}^n$， 和 ${\mathrm{电阻}}_n$ 是一个可接受的函数 ${\mathbb{电阻}}^1$. 各向同性空间过程$X$ 有一个有界的依赖范围，如果 $超\{\tau ：{\mathrm{电阻}}_n\left(\tau \right)\ne 0\}$ $<\infty$. 这里我们考虑一类各向同性的 cf${\mathrm{电阻}}_n\left(\tau \right)$, $n=1,2,\dots$ 有界的依赖范围，其中有 ${\mathrm{电阻}}_1$, 实线上的经典三角形 cf ($n=1$）， 和 ${\mathrm{电阻}}_3$, 球面 cf 在维度三 ($n=3$）。对于每个维度$n\ge 1$, 的可受理性 ${\mathrm{电阻}}_n\left(\tau \right)$作为研究更高维度的 cf。虽然众所周知，对于每个$n\ge 1$, ${\mathrm{电阻}}_n$ 是合法的 cf ${\mathbb{电阻}}^{米}$ 对全部 $米\le n$ 但它表明所考虑的 ${\mathrm{电阻}}_n$ 不是合法的 cf ${\mathbb{电阻}}^{米}$ 什么时候 $米>n$. 因此球形 cf${\mathrm{电阻}}_3$ 不能是比照 ${\mathbb{电阻}}^n$ 什么时候 $n>3$. 各向同性 cf 的识别问题${\mathbb{电阻}}^n$ 讨论了构建各向同性 cf 的简单程序 ${\mathbb{电阻}}^n$ 对于每个 $n\ge 1$给出。这篇文章再次提醒我们必须谨慎对待所选 cf 在相应空间维度中的合法性。