Positive-definite function on a group

A positive-definite function on a group is a mathematical concept that arises in the context of representation theory, harmonic analysis, and probability theory. Specifically, a function defined on a group is called positive-definite if it satisfies certain properties related to sums and inner products. Formally, let \( G \) be a group, and let \( f: G \to \mathbb{C} \) (or \( \mathbb{R} \)) be a function.