The Fourier transform decomposes functions on \(\mathbb R^d\) into frequencies. It turns translation into modulation, convolution into multiplication, and differentiation into polynomial weights.
Fourier transform on \(L^1\)
Integrable functions have bounded continuous transforms.
Definition. For \(f\in L^1(\mathbb R^d)\), define \(\widehat f(\xi)=\int_{\mathbb R^d}e^{-ix\cdot\xi}f(x)\,\mathrm{d}x\).
Proposition. \(\widehat f\) is bounded and uniformly continuous, with \(\|\widehat f\|_\infty\le\|f\|_1\).
Proof. The bound is immediate from \(|e^{-ix\cdot\xi}|=1\). Moreover, \[ \sup_{\xi\in\mathbb R^d}|\widehat f(\xi+h)-\widehat f(\xi)| \le\int|e^{-ix\cdot h}-1|\,|f(x)|\,\mathrm{d}x. \] The integrand tends pointwise to zero and is bounded by \(2|f|\). Dominated convergence therefore proves uniform continuity, not merely pointwise continuity.
Theorem (Riemann-Lebesgue). If \(f\in L^1(\mathbb R^d)\), then \[ \widehat f(\xi)\longrightarrow0 \qquad(\|\xi\|\to\infty). \]
Proof. For \(g\in C_c^1\), integration by parts in a coordinate \(j\) for which \(|\xi_j|\ge\|\xi\|/\sqrt d\) gives \[ |\widehat g(\xi)| \le\frac{\|\partial_jg\|_1}{|\xi_j|} \le\frac{\sqrt d\,\max_k\|\partial_kg\|_1}{\|\xi\|}\longrightarrow0. \] Since \(C_c^1\) is dense in \(L^1\) and \[ \|\widehat f-\widehat g\|_\infty\le\|f-g\|_1, \] the result extends to every \(f\in L^1\).
Convolution theorem
Fourier transform turns convolution into multiplication.
Theorem. If \(f,g\in L^1(\mathbb R^d)\), then \(\widehat{f*g}=\widehat f\,\widehat g\).
Proof. Use Fubini: \[ \widehat{f*g}(\xi)=\int e^{-ix\cdot\xi}\int f(x-y)g(y)\,\mathrm{d}y\,\mathrm{d}x. \] Set \(u=x-y\). The integral factors into \(\int e^{-iu\cdot\xi}f(u)\,\mathrm{d}u\int e^{-iy\cdot\xi}g(y)\,\mathrm{d}y\).
Inversion on Schwartz functions
For rapidly decreasing smooth functions, inversion is exact.
Theorem. If \(f\) is a Schwartz function, then \[ f(x)=(2\pi)^{-d}\int_{\mathbb R^d}e^{ix\cdot\xi}\widehat f(\xi)\,\mathrm{d}\xi. \]
Proof. Let \[ G_\varepsilon(x)=(4\pi\varepsilon)^{-d/2}e^{-|x|^2/(4\varepsilon)}, \qquad \widehat G_\varepsilon(\xi)=e^{-\varepsilon|\xi|^2}. \] Fubini, justified by rapid decay, gives \[ \begin{aligned} (f*G_\varepsilon)(x) &=(2\pi)^{-d}\int e^{ix\cdot\xi} e^{-\varepsilon|\xi|^2}\widehat f(\xi)\,\mathrm{d}\xi. \end{aligned} \] The Gaussian kernels form an approximate identity, so the left side converges uniformly to \(f(x)\). On the right, \(\widehat f\) is Schwartz and hence integrable; dominated convergence removes the Gaussian factor and gives the inversion formula.
Plancherel theorem
Fourier transform extends from \(L^1\cap L^2\) to a unitary operator on \(L^2\).
Theorem. With the standard normalization, \(\|\widehat f\|_2=(2\pi)^{d/2}\|f\|_2\) for \(f\in L^2\), after defining the transform by density.
Proof. For Schwartz \(f,g\), inversion and Fubini give \[ \int_{\mathbb R^d}f(x)\overline{g(x)}\,\mathrm{d}x =(2\pi)^{-d}\int_{\mathbb R^d} \widehat f(\xi)\overline{\widehat g(\xi)}\,\mathrm{d}\xi. \] Taking \(g=f\) proves \[ \|\widehat f\|_2=(2\pi)^{d/2}\|f\|_2. \] Schwartz functions are dense in \(L^2\), so this scaled isometry extends uniquely to all of \(L^2\). Applying the same construction to the inverse transform shows that the extension is onto and hence unitary after normalization.
Differentiation and decay
Smoothness in one domain corresponds to decay in the other.
Proposition. For Schwartz \(f\), \(\widehat{\partial^\alpha f}(\xi)=(i\xi)^\alpha\widehat f(\xi)\) and \(\widehat{x^\alpha f}=i^{|\alpha|}\partial^\alpha_\xi\widehat f\), up to the chosen sign convention.
Proof. The derivative identity follows by integration by parts, with boundary terms vanishing because \(f\) is rapidly decreasing. The multiplication identity follows by differentiating under the integral sign with respect to \(\xi\).
Corollary (Sobolev energy in frequency). For a Schwartz function, \[ \|\partial^\alpha f\|_2^2 =(2\pi)^{-d}\int_{\mathbb R^d} |\xi^\alpha|^2|\widehat f(\xi)|^2\,\mathrm{d}\xi. \] Thus square-integrable derivatives are equivalent to polynomially weighted Fourier energy.
Example (Gaussian self-similarity). With the present convention, \[ f(x)=e^{-|x|^2/2} \quad\Longrightarrow\quad \widehat f(\xi)=(2\pi)^{d/2}e^{-|\xi|^2/2}. \] The Gaussian is the basic shape preserved by Fourier transform; scaling in space produces reciprocal scaling in frequency.