Are Fourier Series of great importance?
-----Xiaofeng Xu (Y13)
Abstract: This essay is to give you a general idea about how Fourier series matters in applications in different areas of science and mathematics. A brief historical background is given at the first part of the essay followed by the basic concepts of Fourier series. After that we will have a look at two real application examples of Fourier series and witness its magic power. Finally, in order to show you the fantastic world opened up by Fourier series, broader applications of Fourier series will be mentioned and introduced very briefly. Therefore, naturally and logically, all these gives the answer to the essay question that Fourier series is definitely important. In 1807, when the study of trigonometric series still remained confusing most of the mathematicians, Joseph Fourier, a young French mathematician presented an astonishing dissertation to the Academy of Science in Paris. In his article, Fourier derived the famous heat equation and asserted that any arbitrary function could be written in the form of trigonometric series. However this bald statement was immediately denied by Joseph-Louis Lagrange who was the grand master of French mathematics at that time. Lagrange asserted in a very strong terms, saying that Fourier’s claim was simply impossible. Meanwhile, Fourier’s idea was not accepted by other mathematicians so his brilliant paper work was not published at that time. Then, in 1822, after 15 years’ of working on heat equation and trigonometric series, much more mature theorems had been established. Fourier published his work in book form, Théorie analytique de la chaleur . It was the publishing of this book that marked the new page of trigonometric series and made Fourier’s work start to be accepted and further researched by other mathematicians. Later on, Fourier’s method and his totally new perspective were adopted by Poisson Dirichlet, Riemann and many other great mathematicians, leading to a more and more mature system based on Fourier’s work. History:
Background knowledge:
We define function f(x) following: The function f(x) is defined in the interval (-L,L) and outside of the interval it is defined as f(x+2L)=f(x) . So this is a periodic function with a period of 2L Here I introduce two types of notations of Fourier series for such a function satisfying all the conditions above.
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