Veronique Fischer - Fourier Analysis

Fourier Analysis

This page accompanies my course in Fourier analysis at the University of Bath. It brings together the lecture notes and exercise sheets, alongside a series of audio companions, The World in Waves.

Course materials

Lecture notes: draft of 22/09/2026

Audio companions - The World in Waves

The World in Waves is a series of audio companions to the course, exploring the history, motivation, applications and broader ideas around Fourier analysis. They're meant to complement the mathematical development in the lectures and notes, not replace it.

🎧 1. The Great String Controversy and Functions

The eighteenth-century argument between d'Alembert, Euler and Daniel Bernoulli over the vibrating string, and what it meant for mathematicians to disagree about what actually counts as a function.

Listen - 27:32

🎧 2. From the Guillotine to the Fourier Transform

The turbulent life of Joseph Fourier - revolutionary politics, Napoleon's Egypt expedition, a brush with the guillotine - set against the mathematical breakthrough in his Analytical Theory of Heat that gave the transform his name.

Listen - 30:03

🎧 3. Why smooth waves overshoot sharp corners

The Gibbs phenomenon: why truncated Fourier series show a characteristic persistent overshoot near a jump discontinuity, no matter how many terms you add - and what this tells us about approximating sharp corners with smooth waves.

Listen - 28:35

🎧 4. The Battle to Legalize Dirac's Spike

How physicist Paul Dirac's delta function - infinitely tall, infinitely thin, integrating to one - challenged what mathematicians thought a function could even be, and the decades-long effort, through Laurent Schwartz's theory of distributions, to make it rigorous.

Listen - 33:36

🎧 5. How your ears perform Fourier analysis

Georg Simon Ohm's acoustic law and Hermann von Helmholtz's studies of tone and timbre, and the question of whether the human ear is itself doing a kind of Fourier analysis on the sounds it hears.

Listen - 36:35

🎧 6. Quantum Uncertainty Is Not a Measurement Error

How de Broglie's matter waves and Schrodinger's wave mechanics made the Fourier transform the mathematical link between position and momentum - and why Heisenberg's uncertainty principle is a structural fact about waves, not a story about clumsy measurement.

Listen - 32:53

🎧 7. Why Symmetry Dictates Physical Reality

How Fourier analysis is really harmonic analysis on the symmetries of translation, and how generalising that idea - through Haar measure, the Peter-Weyl theorem and group representations - extends Fourier's toolkit far beyond sines and cosines.

Listen - 30:27

🎧 8. The Math That Turns Reality Into Bits

The tangled priority story behind the sampling theorem - Whittaker, Nyquist, Kotelnikov and Shannon - and how it lets a suitably band-limited continuous signal, sampled densely enough, be captured exactly by a stream of discrete digital samples.

Listen - 37:19

🎧 9. The 1963 doodle that saved modern computing

How Cooley and Tukey's 1965 fast Fourier transform algorithm - with a forgotten precursor from Gauss around 1805 - turned an impossibly slow calculation into one fast enough to underpin modern computing.

Listen - 30:59

🎧 10. Kakeya, Fourier analysis and a 2026 Fields Medal

The century-old Kakeya needle problem, asking how little space a needle needs to point in every direction, turns out to control how concentrated a Fourier transform can be - and how Hong Wang and Joshua Zahl's 2025 proof of the three-dimensional Kakeya conjecture earned a 2026 Fields Medal.

Listen - 28:09

🎧 11. Why signals need wavelets not spoons

Why ordinary Fourier analysis struggles to say not just which frequencies are present but when they occur, and how Gabor, Morlet, Meyer, Mallat and Daubechies built wavelets to localise both time and frequency at once.

Listen - 37:02

🎧12. How mercury solved the phase problem

How X-ray crystallography, CT scanning, MRI and radio interferometry all reconstruct invisible structure from Fourier-type data, and how the crystallographic "phase problem" was cracked using heavy atoms like mercury - closing the series where it began, with Fourier's hidden frequencies revealing a hidden world.

Listen - 24:57

About the audio companions

The audio companions were developed with the help of generative-AI tools, using sources and themes that I selected. I have reviewed and edited the material and take responsibility for its mathematical content and editorial choices.

They are intended primarily as informal introductions to the history, motivation and broader ideas surrounding Fourier analysis, rather than as rigorous mathematical exposition. Mathematical terms are sometimes used in their everyday or intuitive sense — for example, “smooth” should not necessarily be understood in its precise mathematical meaning. For definitions, statements and proofs, please refer to the lecture notes.