Fourier Transform

⚡ Key Takeaways

The Fourier Transform decomposes a function of time (or space) into the frequencies that compose it. Result: any sufficiently well-behaved function $f(t)$ can be written as a continuous superposition of complex exponentials $e^{i\omega t}$ , with weights $\hat f(\omega)$ . Applies across signal processing, PDEs, quantum mechanics, optics, and probability. Introduced by Joseph Fourier (1807–1822) for heat conduction; generalized through the 19th–20th centuries.

💡 The Core Insight

Complex exponentials are the eigenfunctions of differentiation and translation. Because $\frac{d}{dt}e^{i\omega t} = i\omega\, e^{i\omega t}$ , expressing any function as a sum of these exponentials diagonalizes calculus operators. The Fourier Transform is simply the change of basis that performs this diagonalization, replacing the operator $\frac{d}{dt}$ with multiplication by $i\omega$ .

DefinitionFourier Transform and its Inverse

For an absolutely integrable function $f\in L^1(\mathbb{R})$ , the Fourier Transform $\hat f = \mathcal F\{f\}$ is

1

\quad \hat f(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-i\omega t}\,dt

Derivation hint: project $f$ onto the frequency mode $e^{i\omega t}$ via the (continuous) inner product $\langle f, e^{i\omega t}\rangle$ ; the negative sign comes from the complex conjugate in the inner product. The Inverse Fourier Transform reconstructs $f$ :

2

\quad f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat f(\omega)\, e^{i\omega t}\,d\omega

Derivation hint: substitute [1] into the right side and use $\frac{1}{2\pi}\int e^{i\omega(t-s)}d\omega = \delta(t-s)$ , the Fourier representation of the Dirac delta.

Symbols:

$t$ — time (or space) variable, units e.g. seconds.
$\omega$ — angular frequency, units radians/second; sometimes $\omega = 2\pi\nu$ where $\nu$ is ordinary frequency in Hz.
$i$ — imaginary unit, $i^2=-1$ .
$\hat f(\omega)$ — complex amplitude (magnitude = strength, phase = timing) of frequency $\omega$ .
The factor $\frac{1}{2\pi}$ is a normalization convention; other conventions split it as $\frac{1}{\sqrt{2\pi}}$ on each transform.

⚠️ Convention Dependence

The placement of $2\pi$ and the sign in the exponent vary by textbook and field. Three common conventions: $\hat f(\omega)=\int f e^{-i\omega t}dt$ with $\frac{1}{2\pi}$ on the inverse (used here); the symmetric $\frac{1}{\sqrt{2\pi}}$ version; and the "ordinary frequency" form $\hat f(\nu)=\int f e^{-2\pi i\nu t}dt$ with no prefactor at all. Always check the convention before quoting a formula.

TheoremExistence and Inversion

If $f\in L^1(\mathbb{R})$ then $\hat f$ exists, is bounded ( $|\hat f(\omega)|\le \|f\|_1$ ), is continuous, and vanishes at infinity (Riemann–Lebesgue lemma). If in addition $f$ and $\hat f$ are both in $L^1$ , the inversion formula [2] holds at every point of continuity of $f$ . On $L^2(\mathbb{R})$ the transform extends to a unitary isomorphism (Plancherel's theorem).

TheoremPlancherel / Parseval

The Fourier Transform preserves energy up to a constant:

3

\quad \int_{-\infty}^{\infty} |f(t)|^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty} |\hat f(\omega)|^2\,d\omega

Derivation hint: write $\int f\bar g\,dt$ , insert the inverse transform for $\bar g$ , swap order of integration, and apply the delta-function identity; set $g=f$ .

ProofInversion via the Delta Function

We show that applying $\mathcal F$ then the inverse formula recovers $f$ .

Step 1. Start from the candidate reconstruction and substitute definition [1]:

4

\quad \frac{1}{2\pi}\int_{-\infty}^{\infty}\hat f(\omega)e^{i\omega t}\,d\omega = \frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty} f(s)e^{-i\omega s}\,ds\right)e^{i\omega t}\,d\omega

Derivation hint: just plug [1] into [2], renaming the dummy integration variable $t\to s$ inside.

Step 2. Interchange the order of integration (justified for $f\in L^1\cap L^2$ via Fubini after regularization):

5

\quad = \int_{-\infty}^{\infty} f(s)\left(\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i\omega(t-s)}\,d\omega\right)ds

Derivation hint: collect the $\omega$ -dependent factors $e^{-i\omega s}e^{i\omega t}=e^{i\omega(t-s)}$ and pull $f(s)$ out of the inner integral.

Step 3. Recognize the inner integral as the Dirac delta:

6

\quad \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i\omega(t-s)}\,d\omega = \delta(t-s)

Derivation hint: this is the Fourier representation of $\delta$ ; rigorously interpret it as a limit of Gaussians $\frac{1}{2\pi}\int e^{i\omega u - \epsilon\omega^2}d\omega = \frac{1}{\sqrt{4\pi\epsilon}}e^{-u^2/4\epsilon}\to\delta(u)$ as $\epsilon\to0^+$ .

Step 4. The sifting property of $\delta$ collapses the integral:

7

\quad \int_{-\infty}^{\infty} f(s)\,\delta(t-s)\,ds = f(t)

Derivation hint: $\delta(t-s)$ is zero except at $s=t$ , where it picks out $f(t)$ exactly. Thus inversion holds. $\blacksquare$

∎

JF

Joseph Fourier

1768–1830

Jean-Baptiste Joseph Fourier accompanied Napoleon to Egypt, governed the Isère department, and in his 1822 Théorie analytique de la chaleur argued — against fierce skepticism from Lagrange and Laplace — that an arbitrary function could be represented as a sum of sines and cosines. His claim that even discontinuous functions had such expansions was not fully rigorous but proved revolutionary, forcing mathematicians to sharpen the very definitions of "function," "integral," and "convergence" for the next century.

🧠 Intuition

Think of a musical chord recorded as a single wiggling pressure curve $f(t)$ . Your ear hears distinct notes; the Fourier Transform is the mathematical "ear" that reads off the amplitude and phase of each pure tone $e^{i\omega t}$ . The integral [1] works like a resonance detector: multiplying $f(t)$ by $e^{-i\omega t}$ and integrating gives a large result only when $f$ actually contains oscillation at frequency $\omega$ — at other frequencies the product oscillates and averages to near zero. The magnitude $|\hat f(\omega)|$ tells you "how loud" each frequency is; the phase $\arg\hat f(\omega)$ tells you "when" it peaks.

Figure

A time-domain Gaussian pulse and its frequency-domain transform, which is also a Gaussian — narrower in time means wider in frequency.

ExampleRectangular Pulse → Sinc

Given: $f(t)=1$ for $|t|\le T$ and $0$ otherwise, with $T=2$ s.

Step 1. Apply definition [1]:

8

\quad \hat f(\omega)=\int_{-T}^{T} e^{-i\omega t}\,dt = \left[\frac{e^{-i\omega t}}{-i\omega}\right]_{-T}^{T}

Derivation hint: the integrand is a single exponential; integrate directly over the finite support.

Step 2. Evaluate the bracket:

9

\quad \hat f(\omega)=\frac{e^{i\omega T}-e^{-i\omega T}}{i\omega}=\frac{2\sin(\omega T)}{\omega}

Derivation hint: use $e^{i\theta}-e^{-i\theta}=2i\sin\theta$ and cancel the $i$ .

Step 3. Substitute $T=2$ :

10

\quad \hat f(\omega)=\frac{2\sin(2\omega)}{\omega}=4\,\operatorname{sinc}\!\left(\frac{2\omega}{\pi}\right)

Answer:

\boxed{\hat f(\omega)=\dfrac{2\sin(2\omega)}{\omega}\ \text{(units: s)},\quad \hat f(0)=2T=4\ \text{s}}

The DC value $\hat f(0)=4$ equals the pulse's total area, as expected.

ExampleDecaying Exponential → Lorentzian

Given: $f(t)=e^{-at}$ for $t\ge0$ , $f(t)=0$ for $t<0$ , with decay rate $a=3\ \text{s}^{-1}$ .

Step 1. Apply [1] over the support $[0,\infty)$ :

11

\quad \hat f(\omega)=\int_0^{\infty} e^{-at}e^{-i\omega t}\,dt=\int_0^\infty e^{-(a+i\omega)t}\,dt

Derivation hint: combine exponents into a single complex rate $a+i\omega$ .

Step 2. Integrate (converges since $a>0$ kills the upper limit):

12

\quad \hat f(\omega)=\left[\frac{e^{-(a+i\omega)t}}{-(a+i\omega)}\right]_0^\infty=\frac{1}{a+i\omega}

Derivation hint: at $t=\infty$ the exponential vanishes; at $t=0$ it equals 1.

Step 3. Power spectrum (Lorentzian) at $a=3$ :

13

\quad |\hat f(\omega)|^2=\frac{1}{a^2+\omega^2}=\frac{1}{9+\omega^2}

Answer:

\boxed{\hat f(\omega)=\dfrac{1}{3+i\omega}\ \text{(units: s)},\quad |\hat f(\omega)|^2=\dfrac{1}{9+\omega^2}}

The half-width of the spectral peak equals the decay rate $a=3\ \text{s}^{-1}$ — fast decay gives a broad line.

⚠️ Frequent Errors

"The transform always exists." False for functions not in $L^1$ or $L^2$ in the classical sense; constants, $\sin\omega_0 t$ , and $\delta(t)$ require the distributional (tempered) Fourier Transform.
"Magnitude is all that matters." The phase carries crucial information — swapping the phases of two images (keeping magnitudes) makes each picture look like the other. Phase encodes where features sit.
" $\mathcal F$ is its own inverse." Not quite: $\mathcal F^2 f(t)=2\pi f(-t)$ (a reflection and scaling), so $\mathcal F^4 = (2\pi)^2\,\mathrm{Id}$ . Eigenvalues of $\mathcal F$ are $\{1,i,-1,-i\}$ times constants.
Forgetting the $2\pi$ when mixing conventions corrupts Parseval-type computations.

CounterexampleA Function With No Classical Transform

The constant function $f(t)=1$ is not in $L^1(\mathbb{R})$ , so integral [1] $\int_{-\infty}^{\infty} e^{-i\omega t}\,dt$ does not converge in the ordinary sense. Yet within tempered distributions one finds $\hat 1(\omega)=2\pi\,\delta(\omega)$ : a constant in time is "all DC," a single spike at $\omega=0$ . This shows the naïve integral definition is insufficient and motivates the distributional extension.

connection

The Fourier Transform sits at the center of a web of related tools. For periodic functions it specializes to the Fourier Series, where the continuous integral becomes a discrete sum over harmonics. For sampled, finite data it becomes the Discrete Fourier Transform, computed efficiently by the Fast Fourier Transform. Generalizing the exponent $e^{-i\omega t}$ to a full complex frequency $e^{-st}$ yields the Laplace Transform, better suited to initial-value problems with growth. The transform's deep structure — a unitary change of basis on $L^2$ — connects it to Hilbert Spaces and the Spectral Theorem. The convolution theorem links it to Convolution and to Probability Theory via characteristic functions.

🔬 Where this appears

⏳ History

- 1753 — Daniel Bernoulli proposes trigonometric series solutions to the vibrating string, sparking debate with Euler and d'Alembert over which functions they represent.

- 1807 — Fourier submits his memoir on heat to the French Academy, claiming arbitrary functions expand in sine/cosine series; rejected for lack of rigor.

- 1822 — Fourier publishes Théorie analytique de la chaleur, establishing the method despite ongoing skepticism.

- 1829 — Dirichlet gives the first rigorous convergence conditions for Fourier series.

- 1910 — Plancherel proves the

L^2

isometry theorem, securing the transform on a Hilbert space.

- 1933 — Wiener and others develop the modern

L^2

and generalized-function frameworks.

- 1947 — Laurent Schwartz formalizes tempered distributions, giving rigorous meaning to

\hat\delta

,

\hat 1

, etc.

- 1965 — Cooley and Tukey publish the Fast Fourier Transform, reducing cost from

O(N^2)

to

O(N\log N)

and igniting digital signal processing.

open-problem · Boundaries and Hard Cases

Pointwise convergence of Fourier series for general continuous functions resisted proof until Carleson's theorem (1966) showed convergence almost everywhere for $L^2$ functions — a notoriously deep result, with the analogous question for $L^1$ answered negatively by Kolmogorov's 1923 everywhere-divergent example. The uncertainty principle sets a hard limit: a function and its transform cannot both be sharply localized ( $\Delta t\,\Delta\omega\ge \tfrac12$ ), with equality only for Gaussians. In applications, the phase retrieval problem — recovering a signal from $|\hat f|$ alone, as in crystallography — is generally ill-posed and remains an active research area. Time-frequency localization is addressed by extensions such as the Wavelet Transform and the short-time Fourier transform.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also