Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.

The term "harmonics" originated from the Ancient Greek word harmonikos, meaning "skilled in music".

[7] One of the major results in the theory of functions on abelian locally compact groups is called Pontryagin duality.

[9] This choice of harmonics enjoys some of the valuable properties of the classical Fourier transform in terms of carrying convolutions to pointwise products or otherwise showing a certain understanding of the underlying group structure.

Many applications of harmonic analysis in science and engineering begin with the idea or hypothesis that a phenomenon or signal is composed of a sum of individual oscillatory components.

For example, in a study of tides, the experimentalist would acquire samples of water depth as a function of time at closely enough spaced intervals to see each oscillation and over a long enough duration that multiple oscillatory periods are likely included.

For example, the top signal at the right is a sound waveform of a bass guitar playing an open string corresponding to an A note with a fundamental frequency of 55 Hz.

The different wave components contributing to the sound can be revealed by applying a mathematical analysis technique known as the Fourier transform, shown in the lower figure.