论文标题
维度在广义空间上行走
Dimension Walks on Generalized Spaces
论文作者
论文摘要
令$ d，k $为积极的整数。我们将广义空间称为$ d $二维领域的笛卡尔产品，$ \ mathbb {s}^d $，带有$ k $ -dimensional Euclidean Space，$ \ Mathbb {r}^k $。我们考虑连续函数的类$ {\ Mathcal p}（\ Mathbb {\ Mathbb {s}^d \ times \ times \ Mathbb {r}^k）$ $φ：[-1,1] \ times [0，\ infty）\ to \ mathbb {r} $ \ times \ mathbb {r}^k \ right）^2 \ to \ mathbb {r} $，定义为$ c \ big（（x，y），（x^{\ prime}，y^{\ prime}），y^{\ prime}） \ big）$，$（x，y），\; （x^{\ prime}，y^{\ prime}）\ in \ mathbb {s}^d \ times \ times \ mathbb {r}^k $，是积极的。我们提出了线性操作员，以允许在广义空间内穿过维度，同时保持积极的确定性。
Let $d,k$ be positive integers. We call generalized spaces the cartesian product of the $d$-dimensional sphere, $\mathbb{S}^d$, with the $k$-dimensional Euclidean space, $\mathbb{R}^k$. We consider the class ${\mathcal P}(\mathbb{S}^d \times \mathbb{R}^k)$ of continuous functions $φ: [-1,1] \times [0,\infty) \to \mathbb{R}$ such that the mapping $C: \left ( \mathbb{S}^d \times\mathbb{R}^k \right )^2 \to \mathbb{R}$, defined as $C \Big ( (x,y),(x^{\prime},y^{\prime})\Big ) = φ\Big ( \cos θ(x,x^{\prime}), \|y-y^{\prime}\| \Big )$, $(x,y), \; (x^{\prime},y^{\prime}) \in \mathbb{S}^d \times \mathbb{R}^k$, is positive definite. We propose linear operators that allow for walks through dimension within generalized spaces while preserving positive definiteness.
