Plancherel’s theorem

0.1 Statement of theorem

Plancherel’s Theorem states that the unitary Fourier transformMathworldPlanetmath of 𝐋1 functions (the Lebesgue-integrable functions ( on n extends to a unitary isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on 𝐋2 (the square-integrable functions).

Thus, the following two fundamental properties hold for the Fourier transform on 𝐋2 functions g:n:

  1. i

    The equalities are as elements of 𝐋2; in terms of pointwise functions, the equalities hold almost everywhere on n.

  2. ii

    The Fourier transform preserves 𝐋2 norms:


0.2 Extension of the Fourier transform to 𝐋2

The extensionPlanetmathPlanetmath of the usual Fourier transform can be described concretely as follows: given a 𝐋2 function g:n, take any sequenceMathworldPlanetmathPlanetmath gk:n of 𝐋1 functions that converge in 𝐋2 to g. The Fourier transforms


are defined as usual, and g can be obtained as the 𝐋2 limit of gk.

In the one-dimensional case, a common sequence of approximating sequences to take is gk=g𝕀[-k,k]; in that case we have


The inverse Fourier transform -1 can be obtained in a similar way to , using approximating functions gk:


0.3 Note on different conventions

Here, we have used the convention for the Fourier transform that ξ denotes “ordinary frequency”, i.e. the exponential contains factors of 2π. Another common convention has ξ replaced by ω denoting the “angular frequency”, with factors 2π occurring not in the exponent, but as multiplicative constants. In this case property (i) above still holds, but property (ii) will not hold unless the multiplicative constants in front of the forward and inverse Fourier transform are chosen properly.


  • Folland Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
  • Katznelson Yitzhak Katznelson. An Introduction to Harmonic Analysis, second ed. Dover Publications, 1976.
  • Wiki transform ”, Wikipedia, The Free Encyclopedia. Accessed 22 December, 2006.
Title Plancherel’s theorem
Canonical name PlancherelsTheorem
Date of creation 2013-03-22 16:29:00
Last modified on 2013-03-22 16:29:00
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 11
Author stevecheng (10074)
Entry type Theorem
Classification msc 42B10
Classification msc 42A38
Related topic ProofOfSamplingTheorem