Statement of theorem

Plancherel's Theorem states that the unitary Fourier transform of $\Le^1$ functions (the Lebesgue-integrable functions) on $\real^n$ extends to a unitary isomorphism on $\Le^2$ (the square-integrable functions).

Thus, the following two fundamental properties hold for the Fourier transform $\FT$ on $\Le^2$ functions $g \colon \real^n \to \complex$ : -6-JG The equalities are as elements of $\Le^2$ ; in terms of pointwise functions, the equalities hold almost everywhere on $\real^n$ .

1. The Fourier transform preserves $\Le^2$ norms:$$ \int_{\real^n} \abs{\FT g(\xi)}^2 \, d\xi = \norm{ \FT g }_{\Le^2}^2 = \norm{g}_{\Le^2}^2 = \int_{\real^n} \abs{g(x)}^2 \, dx\,.$$

Extension of the Fourier transform to $\Le^2$

The extension $\FT$ of the usual Fourier transform can be described concretely as follows: given a $\Le^2$ function $g\colon \real^n \to \complex$ , take any sequence $g_k \colon \real^n \to \complex$ of $\Le^1$ functions that converge in $\Le^2$ to $g$ . The Fourier transforms$$ \FT g_k (\xi) = \int_{\real^n} g_k(x) \, e^{-2\pi i \xi \cdot x} \, dx\,, \quad \xi \in \real^n$$ are defined as usual, and $\FT g$ can be obtained as the $\Le^2$ limit of $\FT g_k$ .

In the one-dimensional case, a common sequence of approximating sequences to take is $g_k = g \cdot \indc_{[-k,k]}$ ; in that case we have$$ \FT g(\xi) = \lim_{T \to \infty} \int_{-T}^T g(t) \, e^{-2\pi i \xi t} \, dt\,, \quad \xi \in \real\,.$$

The inverse Fourier transform $\FT^{-1}$ can be obtained in a similar way to $\FT$ , using approximating functions $g_k$ :$$ \FT^{-1} g_k (x) = \int_{\real^n} g_k(\xi) \, e^{2\pi i \xi \cdot x} \, dx\,, \quad x \in \real^n\,.$$

Note on different conventions

Here, we have used the convention for the Fourier transform $\FT$ 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 constants in front of the forward and inverse Fourier transform are chosen properly.

Bibliography

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

`` Fourier transform '', Wikipedia, The Free Encyclopedia. Accessed 22 December, 2006.