Consider the cube . Let be a function defined on . For every , the th Fourier coefficient of is defined to be
where for and we denote . The sum
is called the Fourier series of . The basic problem in Fourier analysis is whether one can reconstruct from its Fourier coefficients, and in particular, under what conditions, and in what sense, does the Fourier series of converge to .
One week into the course, we are ready to start applying the structure theory of Hilbert spaces that we developed in the previous two lectures, together with the Stone-Weierstrass Theorem we proved in the introduction, to obtain easily some results in Fourier series.