Strictly positive definite kernels on the circle
The author studies strictly positive definite and strictly conditionally negative definite functions on the circle \(S^1\). Strict positive definiteness, resp. strict conditional negative definiteness, means that strict inequalities hold whenever nontrivial bilinear forms of the corresponding matrices are considered. In \textit{Y. Xu} and \textit{E. W. Cheney} [Proc. Am. Math. Soc. 116, 977-982 (1992; Zbl 0787.43005)] it was shown that in the Schoenberg series representation \(f(t)= \sum^\infty_{k=0} =a_k \cos kt\) of the positive definite \(f\) one has strict positive definiteness provided all \(a_k>0\). Strict positive (and also strict conditional negative) definiteness will depend on the set of integers \(k\) for which \(a_k<0\) and not on the magnitude of the \(a_k\). In the present paper at first the result of Xu and Cheney is improved: Let \(f(t)= \sum_{k\in K} a_k\cos kt\) with \(a_k>0\) for all \(k\in K\) and \(\sum_{k\in K} a_k<\infty\). If \(K\) is a subset of \(\cup^\infty_{k=0} (b_k+ c_kN_k)\), where \(N_k= \{0,1, \dots, k\}\), \(b_k,c_k \in \mathbb{N}\) and \((c_k)\) is a prime sequence, then \(f\) is strictly positive definite. An increasing sequence \((c_k)\) of nonnegative integers is said to be prime if it is not contained in any set of the form \(p_1\mathbb{N} \cup p_2\mathbb{N} \cup\cdots \cup p_n \mathbb{N}\), where \(p_1, p_2, \dots, p_n\) are prime numbers. After that the author derives three independent necessary conditions for subsets \(K\subseteq \mathbb{N}_0\), where the functions \(f(t)\) with series representations over \(K\) as above are strictly positive definite. Each result for strict positive definiteness holds as well for strict conditional negative definiteness.
strictly positive definite and strictly conditionally negative definite functions
scientific article; zbMATH DE number 878078