Schoenberg's Problem on Positive Definite Functions

Abstract

If $n\ge 3$, $q>2$ and $\beta >0$ then the function $\exp (-(|x_1{|}^q+|x_2{|}^q+\cdots +|x_n{|}^q{)}^{\beta /q})$\ 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.