Fourier Transform in $L^1$ and $L^2$

Motivation: Spectral Decomposition of Signals

The Fourier transform turns any signal into a superposition of sines and cosines of various frequencies. For differential equations this is gold: differentiation $\mapsto$ multiplication by frequency; convolution $\mapsto$ pointwise multiplication. Plancherel's theorem: Fourier is a unitary (isometric) operator on $L^2(\mathbb{R})$, "signal energy" = "total power of harmonics".

Definition and Properties

Fourier transform in $L^1(\mathbb{R})$: $ \hat{f}(\xi) = \int_\mathbb{R} f(x) e^{-2\pi i x \xi} dx. $

Boundedness: $|\hat{f}(\xi)| \leq |f|_{L^1}$. Riemann–Lebesgue lemma: $\hat{f}(\xi) \to 0$ as $|\xi| \to \infty$.

Main properties (when $f, f' \in L^1$):

• Translation: $(f(\cdot - a))\ \hat{}\ (\xi) = e^{-2\pi i a \xi} \hat{f}(\xi)$.
• Differentiation: $(f')\ \hat{}\ (\xi) = 2\pi i \xi \cdot \hat{f}(\xi)$. Derivative $\to$ multiplication by frequency!
• Multiplication by $x$: $(x f)\ \hat{} = \dfrac{i}{2\pi} \cdot (\hat{f})'(\xi)$.
• Convolution: $(f * g)\ \hat{} = \hat{f} \cdot \hat{g}$ — convolution theorem.

Inverse: $f(x) = \int \hat{f}(\xi) e^{2\pi i x \xi} d\xi$ (inversion formula, when $\hat{f} \in L^1$).

Plancherel’s Theorem

Theorem: The Fourier transform extends to an isometric isomorphism $\mathcal{F}: L^2(\mathbb{R}) \to L^2(\mathbb{R})$: $|\hat{f}|{L^2} = |f|{L^2}$.

Parseval's identity: $\int |f(x)|^2 dx = \int |\hat{f}(\xi)|^2 d\xi$.

Inner product: $\langle f, g \rangle = \langle \hat{f}, \hat{g} \rangle$ — $\mathcal{F}$ is unitary on $L^2$.

FFT (Cooley–Tukey, 1965): the discrete analog in $O(N \log N)$ instead of $O(N^2)$. The basis of digital signal processing: audio (MP3), video (H.264), radar.

Application to Differential Equations

Heat equation: $\partial_t u = \partial^2_x u$. Fourier in $x$: $\partial_t \hat{u} = -4\pi^2 \xi^2 \hat{u} \Rightarrow \hat{u}(\xi, t) = \hat{u}_0(\xi) \cdot e^{-4\pi^2 \xi^2 t}$. Inverse: $u(x, t) = \int \hat{u}_0(\xi) e^{-4\pi^2 \xi^2 t} e^{2\pi i x \xi} d\xi = u_0 * G(\cdot, t),\ G(x, t) = e^{-x^2 / (4t)}/\sqrt{4\pi t}$.

Numerical Example

Problem: Compute the Fourier transform of the rectangular window $f = \operatorname{rect}$: $f(x) = 1$ for $|x| < 1/2$, $0$ otherwise.

Step 1. $ \hat{f}(\xi) = \int_{-1/2}^{1/2} e^{-2\pi i x \xi} dx = \left[ \frac{e^{-2\pi i x \xi}}{-2\pi i \xi} \right]_{-1/2}^{1/2} \ = \frac{e^{-\pi i \xi} - e^{\pi i \xi}}{-2\pi i \xi} = \frac{2i \sin(\pi \xi)}{2\pi i \xi} = \textbf{sinc}(\xi) = \frac{\sin(\pi \xi)}{\pi\xi}. $

Step 2. $\mathrm{sinc}(0) = 1$ (limit). Zeros: $\xi = \pm1, \pm2, \ldots$ This is the bandwidth of the rectangular filter.

Step 3. Filtering: $y = f * x \Rightarrow \hat{y} = \mathrm{sinc} \cdot \hat{x}$. The rectangular window is a low-pass filter: passes $|\xi| < 1/2$, suppresses high frequencies ($\mathrm{sinc} \to 0$ as $|\xi| \to \infty$).

Step 4. Parseval's identity: $|f|^2_{L^2} = \int_{-1/2}^{1/2} dx = 1$. $|\hat{f}|^2_{L^2} = \int \mathrm{sinc}^2(\xi) d\xi$. Should be: $=1$ ✓ (by Plancherel's theorem).

Step 5. Gaussian function: $g(x) = e^{-\pi x^2}$. $\hat{g}(\xi) = e^{-\pi \xi^2}$. The Gaussian is a eigenfunction of $\mathcal{F}$ with eigenvalue $1$. This explains why Gaussian noise and Gaussian kernel functions are so important: they are invariant under the Fourier transform.

Step 6. Uncertainty principle: $\sigma_x \cdot \sigma_\xi \geq 1/(4\pi)$. For the Gaussian: $\sigma_x = \sigma_\xi = 1/(2\sqrt{\pi}) \Rightarrow \sigma_x \cdot \sigma_\xi = 1/(4\pi)$ — equality! The Gaussian is the optimal function for the uncertainty principle.

Real-World Application

5G telecommunications: OFDM (Orthogonal Frequency Division Multiplexing) — a method where data is split into thousands of parallel frequency subchannels, orthogonal in the Fourier sense. Every smartphone performs an FFT at every data reception and transmission — a direct application of Plancherel's theorem.

Additional Aspects

The Fourier transform $F: f \to \hat{f}(\omega) = \int f(x) e^{-i \omega x} dx$ is an isometry of $L^2(\mathbb{R})$ onto itself (Plancherel's theorem), translates differentiation into multiplication by $i\omega$, and convolution into multiplication. This turns linear differential equations with constant coefficients into algebraic equations. Schwartz distributions (generalized functions) extend the domain of $F$: the Dirac delta-function $\delta$ has $F[\delta] = 1$, the Heaviside step function — $F[H] = \pi\delta(\omega) + 1/(i\omega)$. Without distributions, it is impossible to rigorously describe impulse signals, point sources in physics, fundamental solutions of PDEs. Modern applications include signal processing (Cooley–Tukey FFT algorithm in $O(N \log N)$), compression (JPEG, MP3), solving PDEs by spectral methods, and quantum field theory.

Connection with Other Areas of Mathematics

In the theory of differential equations, the Fourier transform underlies the spectral approach to linear operators. For the Laplace operator on the real line it realizes it as the operator of multiplication by the quadratic frequency function; this is a particular case of the functional calculus for self-adjoint operators in the spirit of Stone’s theorem and von Neumann’s spectral theorem. In problems on the entire real line and on the torus, the transition to the Fourier series or Fourier integral allows for the formulation of results like the Paley–Wiener theorem on the link between compact support and analyticity.

The connection with abstract algebra appears via representation theory. For abelian locally compact groups, the single Fourier transform on $\mathbb{R}$ generalizes to the dual group (Pontryagin, 1934), and Plancherel's theorem becomes a statement about the isometric property of the transform from $L^2$ of the group to $L^2$ of its dual. In the noncommutative case, the analog is the theory of unitary representations and the Plancherel theorem for semisimple Lie groups (Harish-Chandra, Gelfand–Naimark).

In topology and geometric analysis, Fourier tools underpin the proofs of Sobolev estimates and the Gagliardo–Nirenberg inequalities. Stein's work on harmonic analysis shows how $L^p$ estimates for operators defined by symbols in frequency space lead to the regularity of solutions of elliptic and parabolic equations.

In probability theory, the Fourier transform coincides with the characteristic function of a random variable. The central limit theorem in the formulation of Lévy and Lindberg relies on the convergence of such characteristic functions. For Markov processes with independent increments (Lévy processes), the Fourier image gives the symbol of the generator, which is the foundation of the pseudo-differential approach (Kurtz, Jacob).

Numerical methods use discrete analogs in spectral methods for solving partial differential equations (Gottlieb–Orszag). There, differentiation is approximated by multiplying Fourier coefficients, and the stability and convergence of the schemes are analyzed through the spectrum of the corresponding matrices.

Historical Note and Development of the Idea

The idea of decomposing functions into harmonics appears already in J. Fourier's memoir of 1807 on heat conduction, later published in Monographie sur la chaleur (1822). Initially, the discussion concerned trigonometric series, not the integral over a continuous spectrum. The integral form arose from studies of waves and electromagnetism in the works of Kirchhoff, Helmholtz, and Riemann in the second half of the 19th century. Rigorous analytic justification of the behavior of Fourier series is associated with Riemann’s work (1854), with the works of Dirichlet and Jordan on points of discontinuity and regularity. The transition to $L^2$ functions and the formulation of Parseval’s identity as an isometry are due to P. du Bois-Reymond and M. Plancherel, whose theorem on the isometry of the Fourier transform on $L^2(\mathbb{R})$ (1910) became the starting point for modern harmonic analysis.