ABSTRACT
The purpose of this chapter is to acquaint students with some of the most important aspects of the theory and applications of Fourier series.
Fourier analysis is a branch of mathematics that was invented to solve some partial dierential equations modeling certain physical problems. The history of the subject of Fourier series begins with d'Alembert (1747) and Euler (1748) in their analysis of the oscillations of a violin string. The mathematical theory of such vibrations, under certain simplied physical assumptions, comes down to the problem of solving a particular class of partial dierential equations. Their ideas were further advanced by D. Bernoulli (1753) and Lagrange (1759). Fourier's contributions begin in 1807 with his study of the problem of heat ow presented to the Academie des Sciences. He made a serious attempt to show that \arbitrary" function f of period T can be expressed as an innite linear combination of the trigonometric functions sine and cosine of the same period T :
f(t) =
an cos
2n t T
+ bn sin
2n tT
:
Fourier's attempt later turned out to be incorrect. Dirichlet (1829), Riemann (1867) and Lebesgue (1902) and later many other mathematicians made important contributions to the subject of Fourier series.
