Functions operating on positive definite functions
Let \(G\) be a locally compact abelian group and denote by \(P_1 (G)\) the set of all continuous positive definite functions \(\varphi\) on \(G\) with \(\varphi (0) = 1\). A function \(f\) of \(n \geq 1\) variables is said to operate on \(P_1 (G)\) if \(f(\varphi_1, \dots, \varphi_n) \in P_1 (G)\) whenever \(\varphi_1, \dots, \varphi_n \in P_1 (G)\). The author characterizes such operating functions for a Cantor type group \(G\), e.g., for an infinite \(G\) all nontrivial elements of which have order 2. A function \(f\) operates on \(P_1 (G)\) if and only if \(f\) admits the representation \[ f(x) =(1 - \alpha) \sum^\infty_{|k |= 0} a_k x_1^{k_1} \cdot \dots \cdot x_n^{k_n} + \alpha \int y (x)d \mu (y), \] on the hypercube \([- 1,1]^n\), where \(k\) is a multi-index, \(a_k \geq 0\), \(0 \leq \alpha \leq 1\), and \(\mu\) is a probability measure on the set of functions of the form \(y(x) = y_1 (x_1) \cdot \dots \cdot y_n (x_n)\), where \(y_m (x_m) = x_m^{k_m}\), or \(y_m (x_m) = \delta (-1) + \delta (1)\), \(m = 1, \dots, n\). Moreover, if \(G\) is a nondiscrete group, then \(\alpha = 0\).
scientific article; zbMATH DE number 794484
scientific article