Schoenberg's Problem on Positive Definite Functions

Abstract

If $n \ge 3$, $q>2$ and $β> 0$ then the function $\exp(-(|x_1|^q+|x_2|^q+\dots+|x_n|^q)^{β/q})$\ is not positive definite. This result gives an answer to a question posed by I.J.~Schoenberg in 1938. This text is an authorized English translation of the paper published in Russian in Algebra and Analysis 3(1991), \#3, p.78--85.