Criteria of Pólya type for radial positive definite functions (Q2719005)

The real-valued function \(f\) on the \(n\)-dimensional Euclidean space \(\mathbf R^n\) is called positive definite if the matrix with entries \(f(x_i-x_j),\) \(i,j=1,\dots ,k,\) is nonnegative definite for all finite systems \(x_1,\dots ,x_k\) of points in \(\mathbf R^n.\) By \(\Phi_n,\) \(n=1,2,\dots ,\) the class of continuous functions \(\varphi: [0,\infty)\to\mathbf R\) is denoted such that \(\varphi(0)=1\) and its radial extension \(f(x)=\varphi(|x|),\) \(x\in\mathbf R^n,\) is positive definite. The main result of the paper is the following Pólya type criterion of positive definiteness of radial functions. NEWLINENEWLINENEWLINE\textbf{Theorem 1.1.} Let \(\varphi:[0,\infty)\to\mathbf R\) be a continuous function with \(\varphi(0)=1\) and \(\lim\limits_{t\to\infty}\varphi(t)=0.\) Suppose that \(k\) and \(l\) are nonnegative integers, at least one of which is strictly positive. Put \(\eta_1(t)=\left(-{d\over du}\right)^k\varphi (\sqrt u)|_{u=t^2}.\) If there exists an \(\alpha\geq 1/2\) so that \(\eta_2(t)=\left(-{d\over du} \right)^{k+l-1}[-\eta'_1(t^\alpha)]\) is convex for \(t>0,\) then \(\varphi\) is an element of the class \(\Phi_n\) for \(n=1,\dots ,2l+1.\)NEWLINENEWLINENEWLINERelations of this result to previously known analogues of Pólya's criterion for radial functions are discussed in detail. Numerous applications are given, first of all to the so-called Kuttner-Golubov problem, namely, for which \(\alpha,\beta>0\) the function \((1-t^\beta)^\alpha_+\) is an element of \(\Phi_n.\) The list of references seems to be comprehensive.