The Fourier Transform

The transform is the arithmetic backbone of measurement. In magnetic resonance imaging the scanner does not record an image at all — it records samples of $\tilde f(k)$ directly, filling a grid called k-space, and the picture a radiologist reads is the inverse Fourier transform of that raw data; MRI pulse sequences are strategies for traversing k-space efficiently. In X-ray crystallography and cryo-electron microscopy the diffraction amplitudes are the Fourier transform of the electron density, so recovering a molecule's structure is an inversion problem hobbled by the loss of phase. Radio astronomy's aperture synthesis is the same: an interferometer samples the Fourier transform of the sky brightness (van Cittert–Zernike) and transforms back. The Cooley–Tukey fast Fourier transform of 1965 made all of this computable in $O(N\log N)$ rather than $O(N^2)$ operations, and it now runs inside every audio codec, phone modem, and spectrum analyzer. At the frontier, LIGO extracts gravitational-wave chirps from noise by matched filtering in the frequency domain, and Shor's algorithm factors integers with a quantum Fourier transform — the same change of basis on qubit amplitudes.

Think of $f(x)$ as a chord and the transform as a perfect ear: it reports how much of each pure tone $e^{ikx}$ is present. A sharply localized pulse demands many frequencies to build its steep edges, so its spectrum is broad; a slowly undulating signal needs only low frequencies, so its spectrum is narrow. This is not a rule of thumb but an exact reciprocity — squeeze the function in $x$ by a factor and its transform stretches in $k$ by that factor, since $f(ax)$ transforms to $\tilde f(k/a)/|a|$. The first figure makes the reciprocity and the resulting uncertainty bound one picture.

Each plane wave $e^{ikx}$ satisfies $\tfrac{\mathrm{d}}{\mathrm{d}x}e^{ikx}=ik\,e^{ikx}$, so it is an eigenfunction of differentiation with eigenvalue $ik$; decomposing $f$ into these eigenfunctions is precisely diagonalizing the derivative operator, which is why a constant-coefficient differential equation turns algebraic in $k$-space. The forward and inverse transforms are nearly identical — differing only by the exponent's sign and a $2\pi$ — a duality in which position and wavenumber play interchangeable roles. The starkest case is the boxcar and the sinc: a rectangular pulse of half-width $a$ transforms to an oscillating $\sin(ka)/k$ whose first null sits at $k=\pi/a$, so a narrower box throws its spectral zeros farther out. The second figure is that pair, the same relation that governs single-slit diffraction.

Distributions dissolve the remaining worries. A constant, a plane wave, and the Dirac delta are not in $L^1$ or $L^2$, yet on the space of tempered distributions each has a clean transform: the delta maps to a constant and a constant to a delta. That is not a technicality but the transform's natural home, and it lets the completeness relation $\tfrac{1}{2\pi}\int e^{ik(x-x')}\mathrm{d}k=\delta(x-x')$ carry the whole inversion theorem.

The forward transform is the analysis step: it measures the amplitude of the plane wave $e^{ikx}$ inside $f$. We use the physics convention throughout, with all of the $2\pi$ carried by the inverse.

where: $f(x)$ — the function on the real line (position or time domain); $\tilde f(k)$ — its transform, the amplitude density at wavenumber $k$; $k$ — angular wavenumber (or angular frequency $\omega$); $x$ — position or time. The integral projects $f$ onto $e^{ikx}$, exactly as a Fourier coefficient projects onto one harmonic. The inverse is the synthesis step: it rebuilds $f$ by superposing every plane wave weighted by $\tilde f(k)$, with the $1/2\pi$ the continuum analogue of the $1/L$ normalizing a Fourier coefficient.

where: $\tfrac{1}{2\pi}$ — normalization carried entirely by the inverse in this convention. Together (1) and (2) are the Fourier integral pair. Splitting the $2\pi$ symmetrically makes the transform unitary — the form quantum mechanics and mathematics prefer; the physics is identical, only the bookkeeping changes.

where: $1/\sqrt{2\pi}$ — split so forward and inverse are the same operation up to a sign, making $\mathcal F$ unitary on $L^2(\mathbb R)$. The single most useful structural fact: differentiation becomes multiplication, because $\partial_x$ acts on each basis wave $e^{ikx}$ as multiplication by $ik$.

where: $f'$ — $\mathrm{d}f/\mathrm{d}x$, assuming $f\to0$ at $\pm\infty$; $ik$ — the eigenvalue replacing the derivative in $k$-space. This one line is why constant-coefficient ODEs and PDEs become algebra. Convolution — how every linear translation-invariant system acts on its input — becomes a plain product in $k$-space, the convolution theorem developed in aq-mp-convolution-theorem.

where: $f*g$ — the convolution of $f$ and $g$; $\tilde f\,\tilde g$ — their pointwise product. Filtering, smoothing, and Green's-function solutions are this identity read either way. The transform conserves total energy — the $L^2$ norm in $x$ equals that in $k$ up to the convention factor — Plancherel's theorem, the subject of aq-mp-parseval-plancherel-theorems.

where: $|f|^2$ — energy density in position space; $|\tilde f|^2$ — power spectral density in $k$-space. In the symmetric convention (3) the factor disappears and the map is a genuine isometry. Feeding (1) into (2) forces a representation of the Dirac delta as an integral over all plane waves — the completeness relation recalled from aq-mp-dirac-delta-function.

where: $\delta$ — the Dirac delta expressing completeness of the continuous plane-wave basis. It is itself a transform pair: $\widehat{\delta}=1$ and $\widehat{1}=2\pi\delta$. Finally, the reciprocity of the figures is a theorem: the product of the RMS spreads in the two domains is bounded below, with equality only for a Gaussian. With $k=p/\hbar$ this becomes Heisenberg's principle.

where: $\Delta x$ — RMS spread of $|f|^2$ in position; $\Delta k$ — RMS spread of $|\tilde f|^2$ in wavenumber. No function can be sharp in both domains at once.

1 · The rectangular pulse and the sinc (confidence builder)

A gate function $f(x)=1$ for $|x|<a$ and $0$ otherwise, of half-width $a=1$, is Fourier transformed. Find $\tilde f(k)$, its value at $k=0$, and the location of its first zeros, then read off the reciprocity.

Step 1. The transform is an elementary integral because $f$ is constant on its support:

So $\tilde f(k)=2a\,\mathrm{sinc}(ka)$, and with $a=1$ this is $2\sin(k)/k$ — the amber curve in the second figure.

Step 2. At $k=0$ the sinc equals 1, so $\tilde f(0)=2a=\mathbf{2}$. This is no accident: $\tilde f(0)=\int f\,\mathrm{d}x$ is always the area under $f$, here the area $2a$ of the box. The zero-frequency amplitude is the total "DC content" of the signal.

Step 3. The zeros sit where $\sin(ka)=0$ with $k\ne0$, so the first are at $k=\pm\pi/a=\mathbf{\pm\pi\approx\pm3.14}$. Taking the central-lobe half-width $\Delta k\sim\pi/a$ as a bandwidth and the pulse half-width $\Delta x\sim a$, their product $\Delta x\,\Delta k\sim\pi$ is order unity and independent of $a$ — halving the box doubles the bandwidth. A box of half-width a = 1 transforms to 2·sin(k)/k, with area-2 peak and first zeros at k = ±π; narrowing the box spreads the spectrum.

2 · A Gaussian wavepacket and the uncertainty product (fully worked)

A normalized wavepacket has amplitude $f(x)=e^{-x^2/2\sigma^2}$ with $\sigma=2$ (in units of nm). Compute the RMS position width $\Delta x$, the transform, the RMS wavenumber width $\Delta k$, verify $\Delta x\,\Delta k=\tfrac12$, and translate to a momentum spread using $k=p/\hbar$.

Step 1. The position probability density is $|f|^2=e^{-x^2/\sigma^2}$, a Gaussian of variance $\sigma^2/2$. Hence

Step 2. Transform using the Gaussian result with $a=1/2\sigma^2$: $\tilde f(k)=\sqrt{\pi/a}\,e^{-k^2/4a}=\sqrt{2\pi}\,\sigma\,e^{-\sigma^2k^2/2}$. Its modulus squared $|\tilde f|^2\propto e^{-\sigma^2 k^2}$ is a Gaussian of variance $1/2\sigma^2$, so

Step 3. Multiply the two widths:

The $\sigma$ cancels, so every Gaussian sits exactly on the bound (8) regardless of its width — it is the minimum-uncertainty state, confirming the equality case of the third derivation.

Step 4. With $k=p/\hbar$ the spread becomes $\Delta p=\hbar\,\Delta k$, and the bound reads $\Delta x\,\Delta p=\hbar/2$ — Heisenberg saturated. Numerically $\Delta p=(1.055\times10^{-34})(0.354\times10^{9})=\mathbf{3.7\times10^{-26}\ kg\,m/s}$. Δx = 1.41 nm, Δk = 0.354 nm⁻¹, Δx·Δk = 1/2 exactly, giving Δx·Δp = ℏ/2 — a minimum-uncertainty state.

3 · Diffusion by transform: the heat kernel

Solve the heat equation $\partial_t u=D\,\partial_x^2 u$ on the line with a sharp initial spike $u(x,0)=\delta(x)$, and read off how fast the profile spreads.

Step 1. Transform in $x$. Using the derivative rule (4), $\partial_x^2\to-k^2$, so the PDE becomes an ODE at each $k$:

The spike transforms to $\tilde u(k,0)=\widehat\delta=1$, so $\tilde u(k,t)=e^{-Dk^2 t}$ — a Gaussian in $k$ that narrows as $t$ grows.

Step 2. Invert with the Gaussian transform (here $a=Dt$):

This is the heat kernel: a spreading Gaussian of width $\sqrt{2Dt}$, so features diffuse as $\sqrt{t}$, not $t$ — the signature of a random walk.

Step 3. Put in numbers for thermal diffusion in water, $D\approx1.4\times10^{-7}\,\mathrm{m^2/s}$. After $t=1\,\mathrm{s}$ the RMS width is $\sqrt{2Dt}\approx\mathbf{0.53\ mm}$; reaching $5.3\,\mathrm{mm}$ takes a hundredfold longer, $100\,\mathrm{s}$. The delta spike spreads into a Gaussian of width √(2Dt) ≈ 0.53 mm after 1 s — the transform turning a PDE into one line of algebra.

The Fourier transform is where a claim once dismissed as absurd — that any function is a sum of sines — became the working language of physics. Its author solved heat, not waves, and was doubted by the greatest analysts of his age before rigour caught up with his intuition.

Texts. Stone & Goldbart, Mathematics for Physics, Ch. 2 (function spaces) and Ch. 8 (integral transforms) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 20 · Hassani, Mathematical Physics, 2nd ed., Ch. 8 · Byron & Fuller, Mathematics of Classical and Quantum Physics, Ch. 5 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 13 · Lighthill, Introduction to Fourier Analysis and Generalised Functions · Reed & Simon, Methods of Modern Mathematical Physics I, Ch. IX.

Provenance. The series-to-integral limit, the derivative and convolution rules, Plancherel's relation, the Gaussian transform, and the uncertainty bound are standard, verifiable against Stone & Goldbart, Arfken Ch. 20, Hassani Ch. 8, and Riley–Hobson–Bence Ch. 13. A verifier should recompute the Gaussian integral by completing the square, confirm $\widehat{f'}=ik\tilde f$ by parts, reproduce the top-hat sinc $2a\sin(ka)/(ka)$ with zeros at $k=\pi/a$, check the wavepacket product $\Delta x\,\Delta k=\tfrac12$ (independent of $\sigma$), and confirm the heat kernel $1/\sqrt{4\pi Dt}\,e^{-x^2/4Dt}$ by inverse transform. Both conventions are stated explicitly so all numerical factors are unambiguous.

Prerequisites. Fourier Series and Periodic Functions · Convergence of Fourier Series · The Dirac Delta Function. Leads to. Properties of the Fourier Transform · The Convolution Theorem · Parseval's and Plancherel's Theorems · Tempered Distributions and Fourier Transforms · The Laplace Transform · Green's Functions via Integral Transforms.

Ask the tutor: Ask me to transform any specific function, walk through why differentiation becomes $ik$, or show how the uncertainty bound turns into Heisenberg's principle.