A note on functions preserving positive definiteness

TL;DR

This paper characterizes functions that preserve positive definiteness when applied to kernels, providing a comprehensive understanding of their structure and conditions.

Contribution

It offers a complete characterization of functions that maintain positive definiteness of kernels under composition, extending prior partial results.

Findings

Identifies necessary and sufficient conditions for functions to preserve positive definiteness

Provides a full classification of such functions on intervals satisfying natural conditions

Enhances understanding of kernel transformations in functional analysis

Abstract

The aim of the paper is to give a full characterization of functions f from I into the real line R (where I is an interval in R that satisfies certain natural conditions) such that for any I-valued positive definite kernel K defined on an arbitrary set X the kernel formed by composing f with K is positive definite as well.