---
tag: JT3G2NF0VX
kind: concept
name: Modulation space
msc: [42B35, "46E30", 42C15]
sources:
  - url: https://en.wikipedia.org/wiki/Modulation_space
same_as:
  wikidata: Q6889770
  wikipedia: Modulation space
---

## Summary

A modulation space $M^{p,q}(\R^d)$ is a space of tempered distributions defined by the size of their [[short-time Fourier transform|3DHNP5NEWH]].
The exponent $p$ measures integrability in the time variable and $q$ in the frequency variable.
Modulation spaces are the natural function spaces of [[time–frequency analysis|GQFPVXEWK1]].
They are used to study pseudo-differential operators, Gabor expansions and dispersive equations.

## Explanations

### L1 human

A piece of music is described by which notes sound at which times.
The short-time Fourier transform of a signal gives such a description: a number for each time $x$ and each frequency $\omega$.
A modulation space collects the signals whose time–frequency picture has finite size in a mixed norm.
For $p = q = 2$ the size is the usual energy, and the space is $L^2(\R^d)$.
A Gaussian pulse lies in every modulation space, since its time–frequency picture is again a Gaussian bump.