Encyclopedia/1. The Cosmos & The Natural World/1. Mathematics & Formal Systems/03. Calculus & Mathematical Analysis • Curated by Admin Timeline.sg
This timeline traces the development of Fourier analysis and harmonic series from ancient origins to modern applications, highlighting the contributions of key figures such as Pythagoras, Euler, Fourier, Dirichlet, and others. It covers milestones in understanding periodic functions, trigonometric series, and the evolution of Fourier transforms.
Chronological Storyline (41 Milestones)
500 BCE
Pythagoras and the Music of the Spheres
Pythagoras discovers the relationship between musical harmony and simple integer ratios, laying the foundation for the harmonic series. #mathematics #history
Pythagoras and the Music of the Spheres By Unknown author - Photo by Szilas, 2013-03-04, Public domain, https://commons.wikimedia.org/w/index.php?curid=36520519
400 CE
Indian Mathematicians Use Sine Series
Indian mathematicians like Aryabhata develop sine tables and early infinite series approximations, precursors to Fourier series. #mathematics #history
Indian Mathematicians Use Sine Series By Cpjha13 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=49883192
1225 CE
Nasir al-Din al-Tusi and Trigonometry
Persian scholar Nasir al-Din al-Tusi separates trigonometry from astronomy and develops the sine law, advancing harmonic analysis. #mathematics #islamicgoldenage
Nasir al-Din al-Tusi and Trigonometry By Unknown author - scan of stamp 30 May 2006, Public domain, https://commons.wikimedia.org/w/index.php?curid=829461
1545 CE
Gerolamo Cardano and Complex Numbers
Cardano introduces complex numbers in Ars Magna, later essential for Fourier transforms. #mathematics #history
Gerolamo Cardano and Complex Numbers By Unknown author - https://wellcomeimages.org/indexplus/obf_images/ae/d3/8753d7ad74cb086ba79ccfc75e7f.jpg Gallery: https://wellcomeimages.org/indexplus/image/V0001004.html Wellcome Collection gallery (2018-03-29): https://wellcomecollection.org/works/pcd969qu CC-BY-4.0, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=36391939
1614 CE
John Napier Invents Logarithms
Napier's logarithms simplify calculations, aiding later Fourier analysis. #mathematics #history
John Napier Invents Logarithms By Unknown author - https://www.nationalgalleries.org/art-and-artists/3383, Public domain, https://commons.wikimedia.org/w/index.php?curid=154916365
1637 CE
René Descartes and Analytic Geometry
Descartes links algebra and geometry, providing tools for representing periodic functions. #mathematics #history
René Descartes and Analytic Geometry By After Frans Hals - André Hatala [e.a.] (1997) De eeuw van Rembrandt, Bruxelles: Crédit communal de Belgique, ISBN 2-908388-32-4., Public domain, https://commons.wikimedia.org/w/index.php?curid=2774313
1647 CE
John Wallis's Arithmetica Infinitorum
Wallis introduces the concept of infinite series and interpolates the sine function, foreshadowing Fourier series. #mathematics #history
John Wallis's Arithmetica Infinitorum By After Godfrey Kneller - one or more third parties have made copyright claims against Wikimedia Commons in relation to the work from which this is sourced or a purely mechanical reproduction thereof. This may be due to recognition of the "sweat of the brow" doctrine, allowing works to be eligible for protection through skill and labour, and not purely by originality as is the case in the United States (where this website is hosted). These claims may or may not be valid in all jurisdictions. As such, use of this image in the jurisdiction of the claimant or other countries may be regarded as copyright infringement. Please see Commons:When to use the PD-Art tag for more information., Public domain, https://commons.wikimedia.org/w/index.php?curid=6365975
1690 CE
Jacob Bernoulli and the Harmonic Series
Bernoulli studies the harmonic series (1 + 1/2 + 1/3 + ...) and proves its divergence. #mathematics #history
Jacob Bernoulli and the Harmonic Series By Niklaus Bernoulli (1662-1716) - [2] [3], Public domain, https://commons.wikimedia.org/w/index.php?curid=266673
1748 CE
Euler's Formula and Fourier Series
Leonhard Euler publishes Introductio in analysin infinitorum, containing Euler's formula and early Fourier series expansions. #mathematics #history
Euler's Formula and Fourier Series By Jakob Emanuel Handmann - This file was derived from: Leonhard Euler.jpg Edited by: Bammesk Original source: Kunstmuseum Basel, Public domain, https://commons.wikimedia.org/w/index.php?curid=113056351
1753 CE
Daniel Bernoulli and the Vibrating String
Daniel Bernoulli proposes that any vibration of a string can be represented as a sum of sinusoidal harmonics, an early Fourier series. #mathematics #physics
Daniel Bernoulli and the Vibrating String By UnknownEdited by Bammesk - Basel Historical Museum Link: https://www.hmb.ch/en/museums/objects-in-the-collection/details/s/portraet-des-daniel-bernoulli/, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=114825929
1765 CE
Lagrange and the Fourier Series Controversy
Joseph-Louis Lagrange criticizes Bernoulli's harmonic sum idea, sparking debate on series convergence. #mathematics #history
Lagrange and the Fourier Series Controversy By Unknown author - File:Joseph_Louis_Lagrange.jpg, CC0, https://commons.wikimedia.org/w/index.php?curid=155608232
Dec 21, 1807 CE
Fourier Presents His Theory
Joseph Fourier submits his memoir on heat conduction, asserting that any periodic function can be expressed as a sum of sines and cosines. #mathematics #physics
Fourier Presents His Theory By Julien-Léopold Boilly - This file was derived from: Fourier2.jpg Restored by: Bammesk Original source: https://www.gettyimages.com.au/license/169251384 https://wellcomecollection.org/works/b4qh352u, Public domain, https://commons.wikimedia.org/w/index.php?curid=114366437
1822 CE
Publication of Fourier's Analytical Theory of Heat
Fourier publishes his seminal work, formalizing Fourier series and transforms, revolutionizing mathematical physics. #mathematics #physics
1829 CE
Dirichlet's Convergence Theorem
Peter Gustav Lejeune Dirichlet provides the first rigorous proof of convergence for Fourier series under certain conditions. #mathematics #analysis
Dirichlet's Convergence Theorem By Unknown author - Unknown source, Public domain, https://commons.wikimedia.org/w/index.php?curid=90476
1856 CE
Riemann and the Riemann Integral
Bernhard Riemann develops his integral to handle Fourier series, improving analysis of discontinuous functions. #mathematics #analysis
Riemann and the Riemann Integral By Unknown author - http://www.sil.si.edu/digitalcollections/hst/scientific-identity/explore.htm according to the German Wikipedia., Public domain, https://commons.wikimedia.org/w/index.php?curid=27383
1867 CE
Kelvin and the Harmonic Analyzer
Lord Kelvin (William Thomson) invents a mechanical harmonic analyzer to compute Fourier coefficients for tidal prediction. #mathematics #engineering
Kelvin and the Harmonic Analyzer By T. & R. Annan & Sons; restored by Adam Cuerden[1] - National Galleries of Scotland Accession number: PGP 230.1, Public domain, https://commons.wikimedia.org/w/index.php?curid=141105716
1876 CE
Gibbs Phenomenon Observed
J. Willard Gibbs describes the overshoot effect near discontinuities in Fourier series approximations. #mathematics #signalprocessing
1893 CE
Michelson's Harmonic Analyzer
Albert A. Michelson builds a harmonic analyzer using springs and gears to compute Fourier series up to 80 terms. #mathematics #physics
Michelson's Harmonic Analyzer By The original uploader was Bunzil at English Wikipedia. - Photograph is a higher quality version of the public domain image available from AstroLab or Smithsonian Institution. See Wikipedia:WP:Public_domain#Derived_works_and_restorations_of_works_in_the_public_domain, Public domain, https://commons.wikimedia.org/w/index.php?curid=2622010
1902 CE
Lebesgue Integration and Fourier Analysis
Henri Lebesgue introduces his integral, enabling Fourier analysis of a wider class of functions. #mathematics #analysis
Lebesgue Integration and Fourier Analysis By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=336482
1904 CE
Fejér's Theorem on Cesàro Summation
Lipót Fejér proves that Cesàro summation of Fourier series converges uniformly for continuous functions. #mathematics #analysis
Fejér's Theorem on Cesàro Summation By Unknown author - http://www.math.bme.hu/akademia/elhunytakademikusok.html#63, Public domain, https://commons.wikimedia.org/w/index.php?curid=32636978
1915 CE
Carathéodory and Conformal Mapping
Constantin Carathéodory uses Fourier series in complex analysis for boundary behavior. #mathematics #complexanalysis
Carathéodory and Conformal Mapping By File:Caratheodory.JPG: Unknown authorUnknown author derivative work: Pbrks - This file was derived from: Caratheodory.JPG:, Public domain, https://commons.wikimedia.org/w/index.php?curid=120420206
1923 CE
Wiener's Generalized Harmonic Analysis
Norbert Wiener publishes on generalized harmonic analysis and Tauberian theorems. #mathematics #signalprocessing
Wiener's Generalized Harmonic Analysis By Garry Olsh - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=148383228
1932 CE
Paley–Wiener Theorem
Raymond Paley and Norbert Wiener characterize Fourier transforms of compactly supported functions. #mathematics #analysis
1936 CE
Hardy–Littlewood Circle Method
G. H. Hardy and John Edensor Littlewood develop the circle method using Fourier analysis to solve additive number theory problems. #mathematics #numbertheory
1942 CE
FFT Algorithm by Danielson and Lanczos
Gordon Danielson and Cornelius Lanczos develop an early FFT algorithm for computing Fourier series efficiently. #mathematics #computation
FFT Algorithm by Danielson and Lanczos By Yangwenbo99 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=111271197
1948 CE
Shannon's Sampling Theorem
Claude Shannon connects Fourier analysis to information theory, showing signals can be reconstructed from samples. #mathematics #informationtheory
Shannon's Sampling Theorem By Original: Metacomet Vector: Editor at Large - Own work based on: Bandlimited.png, Public domain, https://commons.wikimedia.org/w/index.php?curid=1873176
1952 CE
Zygmund and Trigonometric Series
Antoni Zygmund publishes Trigonometric Series, a comprehensive textbook on Fourier analysis. #mathematics #analysis
Zygmund and Trigonometric Series By Brak danych - Ze zbiorów Narodowego Archiwum Cyfrowego, z serwisu Szukaj w Archiwach, z zespołu Koncern Ilustrowany Kurier Codzienny - Archiwum Ilustracji https://www.szukajwarchiwach.gov.pl/zespol/-/zespol/55702, CC0, https://commons.wikimedia.org/w/index.php?curid=124636608
1955 CE
Lighthill's Fourier Analysis and Generalized Functions
Michael James Lighthill publishes a classic text introducing Fourier analysis with distributions. #mathematics #analysis
1965 CE
Cooley–Tukey FFT Algorithm
James Cooley and John Tukey publish the modern FFT algorithm, revolutionizing digital signal processing. #mathematics #computation
1966 CE
Carleson's Theorem on Almost Everywhere Convergence
Lennart Carleson proves that Fourier series of L² functions converge almost everywhere. #mathematics #analysis
Mallat and Wavelet Theory By Unknown author - originally here, taken from http://en.wikipedia.org/wiki/File:Stephane_Mallat.jpg, Public domain, https://commons.wikimedia.org/w/index.php?curid=8905989
1982 CE
Grossmann–Morlet Wavelet Transform
Alex Grossmann and Jean Morlet develop the continuous wavelet transform, a time-frequency alternative to Fourier analysis. #mathematics #signalprocessing
Grossmann–Morlet Wavelet Transform By DaBler - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=9163358
1986 CE
Daubechies Orthogonal Wavelets
Ingrid Daubechies constructs compactly supported orthogonal wavelets, revolutionizing signal compression and analysis. #mathematics #signalprocessing
The JPEG image compression standard adopts the discrete cosine transform (DCT), a Fourier-related transform, for image coding. #technology #computing
JPEG Standard Uses DCT By AzaToth - File:Felis_silvestris_silvestris_small_gradual_decrease_of_quality.png, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=43107801
1992 CE
Fourier Analysis in Number Theory: Weil Conjectures
Pierre Deligne completes proof of Weil conjectures using etale cohomology and Fourier transforms. #mathematics #numbertheory
1998 CE
Bourgain's Work on Fourier Restriction
Jean Bourgain makes fundamental contributions to the Fourier restriction problem in harmonic analysis. #mathematics #analysis
Bourgain's Work on Fourier Restriction By George Bergman - This image has been extracted from another file, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=74881233
2002 CE
Tao and the Kakeya Conjecture
Terence Tao advances harmonic analysis with work on the Kakeya conjecture and additive combinatorics. #mathematics #analysis
Tao and the Kakeya Conjecture By Institute for Pure & Applied Mathematics - https://www.youtube.com/watch?v=ddTvK9nlquM, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=191601047
2004 CE
Compressive Sensing via Fourier Sampling
Emmanuel Candès, Justin Romberg, and Terence Tao introduce compressive sensing, using Fourier-like measurements for sparse signal recovery. #mathematics #signalprocessing
2010 CE
Donoho and Curvelets
David Donoho develops curvelets, a multiscale system that improves upon Fourier and wavelet representations for edges. #mathematics #signalprocessing
2014 CE
Fourier Analysis on Finite Groups
Persi Diaconis applies Fourier analysis to random walks on groups, influencing combinatorics and probability. #mathematics #probability
Fourier Analysis on Finite Groups By Steve Castillo Photos - S. Holmes-P. Diaconsis_06, Public domain, https://commons.wikimedia.org/w/index.php?curid=158519116
2020 CE
Fourier Neural Operator for PDEs
Researchers develop the Fourier neural operator, using Fourier transforms to solve partial differential equations with deep learning. #mathematics #machinelearning
