Herglotz's theorem

Herglotz's theorem states that every positive definite function defined on integers can be represented as a sum or integral of complex exponentials, essentially linking these functions to frequency components. In simpler terms, if a function exhibits a certain form of symmetry and positivity, it can be broken down into basic oscillations or waves, much like a musical chord is made up of individual notes. This result is fundamental in areas like signal processing and time series analysis, where understanding the underlying frequencies helps in analyzing and reconstructing signals or data patterns.