---
tag: P3T104G564
kind: concept
name: Band-limited function
msc: ["94A20", "42A38", "94A12"]
sources:
  - url: https://en.wikipedia.org/wiki/Bandlimiting
same_as:
  wikidata: Q806263
  wikipedia: Bandlimiting
---

## Summary

A band-limited function is a function on $\R$ whose [[Fourier transform|DPVM34ADZV]] vanishes outside a bounded interval of frequencies $[-B, B]$.
Such functions are very rigid: by the [[Paley–Wiener theorem|THY12GPZ8C]] they extend to entire functions, so they are determined by their values on any interval.
They can be recovered from samples taken at a rate of $2B$ per unit time.
This is the content of the sampling theorem.
See the [[definition|B4TYSQQC38]].

## Explanations

### L1 human

A band-limited signal contains no frequencies above some limit $B$.
Such a signal cannot change too fast, so it is pinned down by its values at evenly spaced times, $2B$ of them per second.
The function $\sin(\pi x)/(\pi x)$ is the basic example: its spectrum is a box on $[-\frac12, \frac12]$.
Audio recordings are filtered to be nearly band-limited before they are sampled.