Fourier series

See also: Fourierseries

English

The first four partial sums of the Fourier series decomposition of a square wave

Etymology

Named after French mathematician and physicist Joseph Fourier (1768-1830), who pioneered research into such series and their application to problems of heat transfer and vibration.

Noun

Fourier series (plural Fourier series)

1. (mathematics, mathematical analysis) Any series resulting from the decomposition of a periodic function into terms involving cosines and sines (or, equivalently, complex exponentials).
   • 1951 [Blackie & Son], R. C. H. Young (translator), Konrad Knopp, Theory and Application of Infinite Series, [1947, Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, 4th Edition], 1990, Dover, Unabridged unaltered reprint of 2nd English Edition, page 493,

     As we have seen (pp. 369—370), the question of the necessary and sufficient conditions under which the Fourier series of an integrable function converges and represents the given function is one which presents very great difficulties.

   • 2001, David Cruz-Uribe (translator), Javier Duoandikoetxea Zuazo, Fourier Analysis, [1995, Javier Duoandikoetxea Zuazo, Análisis de Fourier], American Mathematical Society, page 20,

     There are excellent discussions of Fourier series and integrals in Katznelson [10] and Dym and McKean [4]. The book by R. E. Edwards [5] is an exhaustive study of Fourier series from a more modern perspective.

   • 2006, K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide, 3rd edition, Cambridge University Press, page 415:

     Unlike Taylor series, a Fourier series can describe functions that are not everywhere continuous and/or differentiable.

Derived terms

• Fourier sine series
• Fourier cosine series

Related terms

• Fourier analysis
• Fourier integral
• Fourier transform

Translations

series of cosine and sine functions

• Armenian: Ֆուրիեի շարք (Furiei šarkʻ)
• Chinese:

  Mandarin: 傅立葉級數／傅立叶级数 (Fùlìyè jíshù), 傅里葉級數／傅里叶级数 (Fùlǐyè jíshù)

• Dutch: Fourierreeks
• Finnish: Fourier’n sarja
• French: série de Fourier f
• German: Fourier-Reihe f, Fourierreihe f
• Hungarian: Fourier-sor (hu)
• Italian: serie di Fourier (it) f
• Japanese: フーリエ級数 (フーリエきゅうすう,fūrie kyūsū)
• Kazakh: Фурье қатары (Fure qatary)
• Korean: 푸리에 급수 (purie geupsu)
• Latin: series Fourieriana f
• Russian: ряд Фурье́ m (rjad Furʹjé)

Further reading

• Fourier analysis on Wikipedia.Wikipedia
• Fast Fourier transform on Wikipedia.Wikipedia
• Fourier transform on Wikipedia.Wikipedia