Strictly positive definite kernels on the torus
A necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the multi-dimensional torus is found in the paper. Namely, the following theorem is proved. Theorem. Let \(f\) be a \(2\pi\)-periodic continuous function on \(\mathbb{R}^m\) and let \(T_m\) be the \(m\)-dimensional torus, i.e. \(T_m=[-\pi,\pi)^m\). Assume the kernel \(T_m\times T_m\ni (x,y)\rightarrow f(x-y)\) is positive definite. Then it is strictly positive definite if and only if the set \(\{n\in \mathbb{Z}^m: \widehat{f}(n)>0\}\) intersects all the translations of each subgroup of \(\mathbb{Z}^m\) that has the form \[ (a_1\mathbb{Z}, a_2\mathbb{Z}, \dots, a_m\mathbb{Z}), \;\;\;a_1,a_2,\dots,a_m\in\mathbb{Z}_{+}\setminus \{0\}. \] For the one-dimensional case the analogous result earlier was obtained in the paper: [the second author et al., Comput. Math. Appl. 51, No. 8, 1233--1250 (2006; Zbl 1153.41307)].
scientific article; zbMATH DE number 6798750
scientific article