History of Fourier Analysis
From a controversial paper on heat conduction in 1807 to the algorithm that powers 5G, streaming music, and MRI machines — trace the story of one of mathematics' most transformative ideas.
A BISMUTH project by Bodhin Industries.
Timeline
Fourier Presents His Theory of Heat
Joseph Fourier presents his memoir "Théorie de la propagation de la chaleur" to the Institut de France, claiming that any function can be represented as a sum of sines and cosines. The committee — which included Lagrange, Laplace, and Legendre — is skeptical. Lagrange famously objects, and the paper is not published.
Théorie Analytique de la Chaleur Published
Fourier finally publishes his full treatise on heat conduction. The book introduces the Fourier series and Fourier integral, establishing the idea that periodic functions can be decomposed into sinusoidal components. Despite lingering doubts about mathematical rigor, the work transforms both mathematics and physics.
Dirichlet's Convergence Conditions
Peter Gustav Lejeune Dirichlet provides the first rigorous proof of convergence for Fourier series under specific conditions (the Dirichlet conditions). His work addresses the mathematical objections raised against Fourier's original claims and puts the theory on solid ground.
Riemann Integral & Fourier Analysis
Bernhard Riemann extends Fourier analysis by developing the Riemann integral, providing a more general framework for understanding when and how Fourier series converge. His habilitation thesis investigates the representation of functions by trigonometric series, opening new mathematical directions.
Campbell, Foster & Telephone Filters
George Ashley Campbell and Ronald M. Foster at AT&T develop the theory and practice of electrical filters for telephone networks, applying Fourier analysis to design band-pass filters that allow multiple conversations to share a single wire. Their work launches the field of analog signal processing.
Wiener's Generalized Harmonic Analysis
Norbert Wiener publishes his landmark paper extending Fourier analysis to broad classes of functions, including those that arise in probability and random processes. His work lays the mathematical foundation for signal processing, noise analysis, and what would become known as the Wiener filter.
Shannon's Sampling Theorem
Claude Shannon publishes "Communication in the Presence of Noise," formalizing the Nyquist–Shannon sampling theorem: a band-limited signal can be perfectly reconstructed from samples taken at twice its highest frequency. This theorem becomes the cornerstone of digital audio, digital communications, and all of DSP.
Cooley–Tukey FFT Algorithm
James Cooley and John Tukey publish "An Algorithm for the Machine Calculation of Complex Fourier Series," reducing the computation of the DFT from O(N²) to O(N log N). This single paper makes real-time spectral analysis practical and is widely considered one of the most important algorithms of the 20th century.
The DSP Revolution Begins
Dedicated digital signal processing hardware emerges. Texas Instruments releases early DSP chips, and the first consumer products using FFT-based processing appear — including digital audio equipment, spectrum analyzers, and early voice synthesizers. Universities begin offering DSP courses.
Digital Media & Compression
The JPEG standard (1992) uses the DCT for image compression. MP3 (1993) and AAC use the MDCT for audio compression. The CD, DVD, and digital television all rely on Fourier-based processing. The internet begins carrying compressed audio and video at scale.
Wireless & Medical Breakthroughs
OFDM — built entirely on the FFT — becomes the modulation scheme for WiFi (802.11a/g/n), 4G LTE, and digital broadcasting. MRI scanners achieve unprecedented resolution using fast 3D Fourier reconstruction. FFT-based processing runs in real time on commodity hardware.
AI, 5G & Beyond
5G NR uses massive FFTs (up to 4096 points) for wide-bandwidth channels. Machine learning speech recognition starts with FFT-derived features (spectrograms, MFCCs). GPU-accelerated FFTs enable real-time processing of enormous datasets. The Fourier transform remains as essential as ever — and more accessible than ever.
Key Figures
Joseph Fourier
1768 – 1830French mathematician and physicist who proposed that any function can be represented as a series of sines and cosines. His work on heat conduction, initially rejected by the French Academy, ultimately founded an entirely new branch of mathematics. He also served as a prefect under Napoleon and conducted early research on the greenhouse effect.
Key contribution: The Fourier series and Fourier integral — the conceptual foundation for everything on this site.
Peter Gustav Lejeune Dirichlet
1805 – 1859German mathematician who provided the first rigorous conditions under which a Fourier series converges to the function it represents. His "Dirichlet conditions" — piecewise continuity and a finite number of extrema — settled decades of debate about the validity of Fourier's claims and established the mathematical standards for harmonic analysis.
Key contribution: Rigorous convergence proof for Fourier series (1829).
Claude Shannon
1916 – 2001American mathematician, electrical engineer, and the "father of information theory." Shannon formalized the Nyquist–Shannon sampling theorem, proving that a band-limited continuous signal can be perfectly reconstructed from discrete samples. His 1948 paper "A Mathematical Theory of Communication" also created the field of information theory, defining the fundamental limits of data compression and transmission.
Key contribution: The sampling theorem — the bridge between continuous and discrete signal analysis.
James Cooley & John Tukey
1926–2016 & 1915–2000Cooley (IBM mathematician) and Tukey (Princeton statistician) published the radix-2 FFT algorithm in 1965, reducing the DFT from O(N^2) to O(N \log N) operations. The algorithm was developed partly to detect Soviet nuclear tests from seismic data. Its publication sparked the entire field of digital signal processing and remains the most widely used FFT variant today.
Key contribution: The FFT algorithm — making real-time spectral analysis possible.
The Idea That Changed Everything
Fourier's radical claim: any function — no matter how irregular — can be written as an infinite sum of sines and cosines.
Where the coefficients are:
This deceptively simple idea turned out to be one of the most powerful tools in all of science and engineering. Two hundred years later, we're still finding new ways to use it.
Where We Are Now
Today, FFTs run billions of times per second across the world's computers, smartphones, base stations, and medical devices. Modern implementations take advantage of SIMD instructions, GPU parallelism, and specialized DSP hardware to process enormous datasets in real time.
The mathematics Fourier introduced in 1807 now underpins technologies he could never have imagined: 5G wireless networks, deep-learning speech recognition, gravitational wave detection, and quantum computing simulations.
At Fourier Tools, we believe this powerful mathematics should be accessible to everyone. Our free, browser-based tools let you explore, compute, and visualize Fourier analysis without installing anything.