Plancherel’s theorem

0.1 Statement of theorem

Plancherel’s Theorem states thatthe unitary Fourier transform of ${𝐋}^1$ functions(the Lebesgue-integrable functions (http://planetmath.org/Integral3)) on ${ℝ}^n$extends to a unitary isomorphism on ${𝐋}^2$ (the square-integrable functions).

Thus, the following two fundamental properties holdfor the Fourier transform $ℱ$ on${𝐋}^2$ functions $g:{ℝ}^n\to ℂ$:

1. $${ℱ}^{-1}(ℱg)=g=ℱ({ℱ}^{-1}g).$$

   The equalities are as elements of ${𝐋}^2$; in terms of pointwise functions,the equalities hold almost everywhere on ${ℝ}^n$.

2. The Fourier transform preserves ${𝐋}^2$ norms:

   $${\int }_{{ℝ}^n}{|ℱg(\xi )|}^2𝑑\xi ={\parallel ℱg\parallel }_{{𝐋}^2}^2={\parallel g\parallel }_{{𝐋}^2}^2={\int }_{{ℝ}^n}{|g(x)|}^2𝑑x.$$

0.2 Extension of the Fourier transform to ${𝐋}^2$

The extension $ℱ$ of the usualFourier transform can be described concretely as follows: given a ${𝐋}^2$ function $g:{ℝ}^n\to ℂ$,take any sequence $g_k:{ℝ}^n\to ℂ$of ${𝐋}^1$ functions that converge in ${𝐋}^2$ to $g$.The Fourier transforms

$$ℱg_k(\xi )={\int }_{{ℝ}^n}{g}_k(x)e^{-2\pi i\xi \cdot x}𝑑x,\xi \in {ℝ}^n$$

are defined as usual,and $ℱg$ can be obtained as the ${𝐋}^2$ limitof $ℱg_k$.

In the one-dimensional case, acommon sequence of approximating sequences to take is$g_k=g\cdot {𝕀}_{[-k,k]}$; in that case we have

$$ℱg(\xi )=\underset{T\to \mathrm{\infty }}{lim}{\int }_{-T}^{T}g(t)e^{-2\pi i\xi t}𝑑t,\xi \in ℝ.$$

The inverse Fourier transform ${ℱ}^{-1}$ can be obtained in a similar wayto $ℱ$, using approximating functions $g_k$:

$${ℱ}^{-1}{g}_k(x)={\int }_{{ℝ}^n}{g}_k(\xi )e^{2\pi i\xi \cdot x}𝑑x,x\in {ℝ}^n.$$

0.3 Note on different conventions

Here, we have used the convention for the Fourier transform $ℱ$ that $\xi$ denotes “ordinary frequency”, i.e. the exponential contains factors of $2\pi$. Another common convention has $\xi$ replaced by $\omega$ denoting the “angular frequency”, with factors $2\pi$ 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 constantsin front of the forward and inverse Fourier transform are chosen properly.