10.1 The Fourier transform

For an integrable function \(f\) on \(\mathbb R\), define \[ \hat f(\xi ) = \int _{-\infty }^\infty f(x)e^{-ix\xi }\,dx, \qquad \check g(x) = \frac 1{2\pi }\int _{-\infty }^\infty g(\xi )e^{ix\xi }\,d\xi . \] The basic rules follow from integration by parts and changes of variable: for suitable \(f\) and \(g\),

\(\widehat {f'}(\xi ) = i\xi \hat f(\xi )\) and \(\widehat {xf}(\xi ) = i\hat f'(\xi )\): smoothness of \(f\) is decay of \(\hat f\), and decay of \(f\) is smoothness of \(\hat f\);
\(\widehat {f(\cdot - a)}(\xi ) = e^{-ia\xi }\hat f(\xi )\) and \(\widehat {f(\lambda \,\cdot )}(\xi ) = |\lambda |^{-1}\hat f(\xi /\lambda )\);
\(\widehat {f * g} = \hat f\,\hat g\), where \((f * g)(x) = \int f(x - y)g(y)\,dy\);
\(\widehat {e^{-ax^2}} = \sqrt {\pi /a}\,e^{-\xi ^2/4a}\) and \(\widehat {e^{-a|x|}} = 2a/(a^2 + \xi ^2)\) (Software 10.4.1).

By the scaling rule, a narrow function has a wide transform and conversely; for the Gaussian the product of the widths is fixed, which is the uncertainty principle of quantum mechanics in its simplest form.

Theorem 10.1.1 (Inversion and Plancherel). If \(f\) and \(\hat f\) are both integrable, then \(f = (\hat f)\check {}\) at every point where \(f\) is continuous. If moreover \(f\) is square-integrable, then \[ \int _{-\infty }^\infty |\hat f(\xi )|^2\,d\xi = 2\pi \int _{-\infty }^\infty |f(x)|^2\,dx . \]

Proof. Insert a Gaussian damping factor. By Fubini and the Gaussian transform, \[ \frac 1{2\pi }\int \hat f(\xi )e^{ix\xi }e^{-\varepsilon \xi ^2}\,d\xi = \int f(y)\,\frac 1{2\pi }\int e^{i(x - y)\xi - \varepsilon \xi ^2}\,d\xi \,dy = \int f(y)\,\Phi (x - y, \varepsilon )\,dy , \] where \(\Phi (\cdot , \varepsilon ) = e^{-x^2/4\varepsilon }/\sqrt {4\pi \varepsilon }\) is the heat kernel with \(\kappa = 1\). As \(\varepsilon \to 0\) the left side tends to \((\hat f)\check {}(x)\) by dominated convergence, since \(\hat f\) is integrable, and the right side tends to \(f(x)\) at points of continuity, by the proof of Theorem 6.1.1. For Plancherel, apply inversion to \(f * \tilde f\) with \(\tilde f(x) = \overline {f(-x)}\), whose transform is \(|\hat f|^2\), and evaluate at \(x = 0\): \(\int |f|^2 = (f * \tilde f)(0) = \frac 1{2\pi }\int |\hat f|^2\). □

Plancherel’s theorem is the continuous form of Parseval’s identity, and it allows the transform to be extended by continuity to every square-integrable function. In \(\mathbb R^n\) the same definitions hold with \(x\cdot \xi \) in the exponent and \((2\pi )^{-n}\) in the inverse, and \(\widehat {\partial _jf} = i\xi _j\hat f\), so \(\widehat {\Delta f} = -|\xi |^2\hat f\).

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