This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of strictly positive definite kernel functions is again strictly positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.

翻译：本文关注从有限离散样本集进行多变量逼近的乘积核的构造与特征刻画。为此，我们考虑组合不同分量核，每个分量核作用在低维欧几里得空间上。根据Aronszajn（1950）的研究，半正定核函数的乘积仍为半正定，且其对应的原生空间是张量积的一个特例，称为希尔伯特张量积。我们首先分析乘积核在多变量插值中的一般性问题，然后进一步研究张量积结构，特别是针对网格状样本。利用这一情形，我们证明严格正定核函数的乘积仍为严格正定。此外，我们为著名的牛顿基发展了一套高效计算方案。支持性数值示例展示了乘积核的良好性能，尤其体现在其灵活性上。