On differentiability and analyticity of positive definite functions
The paper deals with positive definite functions \(f:\mathbb{R} \to \mathbb{C}\), that is such that the inequality \(\sum_{j,k=1}^n f(x_j-x_k) \xi_j \overline{\xi_k} \geq 0\) holds for every choice of \(x_1,x_2,\ldots,x_n \in \mathbb{R}\) and \(\xi_1,\xi_2,\ldots,\xi_n \in \mathbb{C}.\) All results of the paper demonstrate the following effects: the global behavior of a smooth positive definite function is determined solely by the sequence of even-order derivatives at the origin. More precisely, the authors prove the following theorems. Theorem 1: Let \(f:\mathbb{R} \to \mathbb{C}\) be a positive definite function and suppose \(f\) is of class \(C^{2n}\) in a neighborhood of the origin for some positive integer \(n.\) If \(f^{(2m)}(0)=0\) for some positive integer \(m \leq n,\) then \(f\) is constant on \(\mathbb{R}\). Theorem 2: Let \(f:\mathbb{R} \to \mathbb{C}\) be a positive definite function and suppose \(f\) is of class \(C^{\infty}\) in a neighborhood of the origin. Then, if there exist positive constants \(M\) and \(D\) such that \[ 0\leq (-1)^n f^{(2n)}(0) \leq D\frac{(2n)!}{M^{2n}} \] for every non-negative integer \(n,\) we have: (i) \(f\) is analytic in \(\mathbb{R};\) (ii) Let \[ 1/h=\limsup \left({\frac{|f^{(2n)}(0)|}{(2n)!}}\right)^{\frac{1}{2n}}. \] There exist \(\alpha, \beta \in [h, +\infty]\) such that \(f\) holomorphically extends to the complex strip \(\{z\in \mathbb{C}: -\alpha < \mathrm{Im}(z) <\beta\},\) where \(\alpha\) and \(\beta\) are maximal with this property. Moreover, if \(h < \infty,\) the function \(f\) cannot be holomorphically extended to both the points \(z=ih\) and \(z=-ih\) simultaneously, implying in particular that \(h=\min\{\alpha,\beta\}\). The proof of these theorems are based on the fact that if a positive definite function \(f\) belongs to the class \(C^{2n}\) in a neighborhood of the origin for some positive integer \(n,\) then \(f \in C^{2n}(\mathbb{R})\) and for all integers \(0\leq m_1\leq n\), \(0\leq m_2\leq n\) and every \(x \in \mathbb{R}\) the inequality \[ |f^{(m_1+m_2)}(x)|^2 \leq (-1)^{(m_1+m_2)}f^{(2m_1)}(0)f^{(2m_2)}(0) \] is valid.
scientific article; zbMATH DE number 5835304
scientific article