---
tag: CMMDQ13YEE
kind: definition
name: Harmonic analysis
msc: [42A99, 42B99, 43A99]
defines: 7QNRC9SJVV
uses: [G12KXDHPPS, DPVM34ADZV]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_analysis
---

## Definition

Let $X$ be a space with a group of symmetries, such as the circle $\mathbb{T} = \R / 2\pi\Z$ or the space $\R^n$ with translations.
*Harmonic analysis* is the branch of analysis that studies functions on such $X$ through the following questions:

1. how a function decomposes into basic oscillations, such as the exponentials $e^{inx}$ on $\mathbb{T}$ or $e^{2\pi i x \cdot \xi}$ on $\R^n$;
2. in which sense, and in which function spaces, the function is recovered from this decomposition;
3. how operators that commute with translations, such as convolutions and differential operators with constant coefficients, act on the decomposition, and which norm estimates they satisfy.

## Remarks

Example: the theory of [[Fourier series|G12KXDHPPS]] answers these questions on $\mathbb{T}$, and the [[Fourier transform|DPVM34ADZV]] answers them on $\R^n$.
Non-example: the study of the integer solutions of a single polynomial equation by elementary congruences decomposes no function into oscillations, so it is not harmonic analysis.
The part over $\R^n$ and $\mathbb{T}^n$ is often called Euclidean or classical harmonic analysis.
The part over locally compact groups is called abstract harmonic analysis, or [[harmonic analysis on groups|5E6KB3V8R0]].