Schoenberg's Problem on Positive Definite Functions

TL;DR

This paper proves that a specific class of exponential functions involving sums of absolute values raised to a power are not positive definite in dimensions three and higher, answering a question posed by Schoenberg in 1938.

Contribution

It provides a definitive answer to Schoenberg's 1938 question by establishing non-positive definiteness of certain exponential functions in higher dimensions.

Findings

Functions of the form exp(-(|x_1|^q + ... + |x_n|^q)^{β/q}) are not positive definite for n ≥ 3, q > 2, β > 0.

The result resolves a long-standing open problem in the theory of positive definite functions.

The paper confirms the limitations of certain exponential functions in higher-dimensional spaces.

Abstract

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