Bochner's theorem

Bochner's theorem characterizes all continuous, positive-definite functions on real Euclidean space as the Fourier transforms of finite, positive measures. In simple terms, it states that any function that behaves like a "generalized correlation" function can be represented as an integral over a spectrum of frequencies, where each frequency is weighted positively. This connection allows us to understand complex functions in terms of their underlying frequency components, bridging the gap between functions and measures. The theorem is fundamental in fields like signal processing, probability, and harmonic analysis.