Positive definite functions on products of metric spaces via generalized Stieltjes functions,Proceedings of the American Mathematical Society

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Positive definite functions on products of metric spaces via generalized Stieltjes functions
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-08-14 , DOI: 10.1090/proc/15137
V. Menegatto

Abstract:For quasi-metric spaces $(X,\rho )$ and $(Y,\sigma )$ and a positive real number $\lambda$ , we propose a model for generating positive definite functions $G_r: \{\rho (x,x'):x,x'\in X\} \times \{\sigma (y,y'):y,y' \in Y\} \mapsto \mathbb{R}$ having the form

$\displaystyle G_r(t,u)=\frac {1}{h(u)^r} f\left (\frac {g(t)}{h(u)}\right ),$

where $r\geq \lambda$ ,

belongs to a convex cone $\mathcal {S}_\lambda ^b$ of bounded completely monotone functions,

is a nonnegative valued conditionally negative definite function on $(X,\rho )$ , and

is a positive valued conditionally negative definite function on $(Y,\sigma )$ . In the case where $(X,\rho )$ and $(Y,\sigma )$ are metric spaces, we determine necessary and sufficient conditions for the strict positive definiteness of the model. The cone $\mathcal {S}_\lambda ^b$ possesses well-established stability properties that allow alternative formulations of the model leading to many classes of positive definite and strictly positive definite functions on $X\times Y$ . If $X=\mathbb{R}^d$ , $Y=\mathbb{R}$ , $\rho$ is the Euclidean distance on

, $\sigma ^{1/2}$ is the Euclidean distance on

, $t\geq 0$ ,

is a positive valued function with a completely monotone derivative, and $\lambda =d/2$ , then $\{G_r:r\geq \lambda \}$ is a subset of the Gneiting's class of covariance space-time functions on $X\times Y$ frequently dealt with in the literature.

[1]

中文翻译：

通过广义Stieltjes函数对度量空间乘积的正定函数

摘要：对于拟度量空间 $（X，\ rho）$ 和一个正实数，我们提出了一个生成正定函数的模型，其形式为 $（Y，\ sigma）$ $\ lambda$ $G_r：\ {\ rho（x，x'）：x，x'\ in X \} \ times \ {\ sigma（y，y'）：y，y'\ in Y \} \ mapsto \ mathbb { R}$

$\ displaystyle G_r（t，u）= \ frac {1} {h（u）^ r} f \ left（\ frac {g（t）} {h（u）} \ right），$

其中，属于有界完全单调函数的凸锥，是上的一个非负值条件负定函数，并且是上的一个正值条件负定函数。在和是度量空间的情况下，我们确定模型的严格正定性的必要条件和充分条件。圆锥体具有完善的稳定性，可以使用该模型的替代公式，从而导致上的许多正定函数和严格正定函数。如果，，是对欧氏距离，是对欧氏距离 $r \ geq \ lambda$

$\ mathcal {S} _ \ lambda ^ b$

$（X，\ rho）$

$（Y，\ sigma）$ $（X，\ rho）$ $（Y，\ sigma）$ $\ mathcal {S} _ \ lambda ^ b$ $X \ Y$ $X = \ mathbb {R} ^ d$ $Y = \ mathbb {R}$ $\ rho$

$\ sigma ^ {1/2}$

，

， $t \ geq 0$ ，

是用完全单调衍生物的正值函数，并且，然后是Gneiting的类的协方差时空功能的上一个子集频繁处理在文献中。 $\ lambda = d / 2$ $\ {G_r：r \ geq \ lambda \}$ $X \ Y$

参考文献[增强功能上关]（这是什么？）