Exponential convexity and total positivity
Let \(I\) be one of the intervals \((0,\infty)\) and \((-\infty,\infty)\) and let \(f:I\rightarrow\mathbb{R}\) be a function associated to a continuous weight \(p:(a,b)\rightarrow\mathbb{R}_{+}\) via the formula \(f(x)=\int_{a}^{b}e^{xt}p(t)\mathrm{d}t.\) The main result of the paper under review is Theorem 2, that asserts that the kernel \(K(x,y)=f(x+y)\) is totally positive on \(I\times I\) (that is, \(\det[f(x_{i}+y_{j})]_{i,j=1}^{n}>0\) for all integers \(n\geq1\) and all families of points \(x_{1}\) \(<\cdots<\) \(x_{n}\) and \(y_{1}<\cdots<y_{n}\) in \(I).\) Several examples illustrating this result are included.
scientific article; zbMATH DE number 7223664