Fourier series

Definitions

Any series resulting from the decomposition of a periodic function into terms involving cosines and sines (or, equivalently, complex exponentials).

Equivalents

Examples

“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.”

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

CEFR level