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10.2we - 10.2 FOURIER SERIES KIAM HEONG KWA Historical note...

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10.2: FOURIER SERIES KIAM HEONG KWA Historical note. On December 21, 1807, an engineer named Joseph Fourier announced to the prestigious French Academy of Sciences that an arbitrary function could be expanded in an infinite series of sines and cosines 1 . His an- nouncement caused a loud furor in the Academy. Many prominent members of the Academy, including the famous mathematician Lagrange, thought this result to be pure nonsense, since at that time it could not be placed on a rigorous foundation due to the lack of a precise notion of function and integral in the early nineteehth century. Later, Dirichlet and Riemann expressed Fourier’s results with greater precision and formality. Although Fourier’s claim of generality was somewhat too strong, his results inspired a flood of important research that has continued to the present day. Just recently, mathematicians have succeeded in establishing exceedingly sharp conditions for the Fourier series to converge. This result ranks as one of the great mathematical theorems of the twentieth century. Our goal in this and the next two sections is to discuss some suf- ficient conditions under which a function f ( x ) can be expanded in a trigonometric series such as (0.1) F [ f ]( x ) = a 0 2 + X n =1 a n cos nπx L + b n sin nπx L , where the coefficients { a n } n =0 and { b n } n =1 are sequences of real num- bers such that F [ f ]( x ) converges to f ( x ) for almost every x in the domain of the latter. Also, we have assumed that f ( x ) is 2 L -periodic. As an intermediary goal, we will calculate the coefficients { a n } n =0 and { b n } n =1 in terms of f ( x ) in this section. Remark 1. If the series F [ f ]( x ) in (0.1) is convergent everywhere, then it has a period of 2 L .
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