I'm having a hard time understand this theorem in our book:(adsbygoogle = window.adsbygoogle || []).push({});

The Plancherel Theorem

The Fourier transform, defined originally on ##L^1\cap L^2## extends uniquely to a map from ##L^2## from ##L^2## to itself that satisfies

##\langle \hat f, \hat g \rangle = 2\pi \langle f,g\rangle## and ##||\hat f||^2= 2\pi||f||^2##

for all ##f,g\in L^2##.

I don't really understand the formulation here. Some questions:

What does it mean "extends uniquely" here?

When am I allowed to use the theorem? Can I use the formula for every ##L^2## function?

Is the Fourier transform of an ##L^2## function always an ##L^2## function as well (even if the function is both in ##L^1## and ##L^2##)?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Plancherel Theorem (Fourier transform)

Loading...

Similar Threads - Plancherel Theorem Fourier | Date |
---|---|

B Some help understanding integrals and calculus in general | May 22, 2017 |

I Noether's theorem | Feb 26, 2017 |

I Proofs of Stokes Theorem without Differential Forms | Jan 24, 2017 |

I Visual interpretation of Fundamental Theorem of Calculus | Dec 27, 2016 |

Plancherel's Identity proof - justifying order of integration? | May 8, 2011 |

**Physics Forums - The Fusion of Science and Community**