Fourier series

In mathematics, a Fourier series, named in honor of Joseph Fourier, is a representation of a periodic function as a sum of periodic functions of the form

which are harmonics of a fundamental. Suppose f(x) is a complex-valued function of a real number, is periodic with period 2π, and is square integrable over the interval from 0 to 2π. Let

Then the Fourier series representation of f(x) is given by

Since

this is equivalent to representing f(x) as a infinite linear combination of functions of the form

, i.e.

Does this series actually converge to f(x)?

A partial answer is that if f is square-integrable then

That much was proved in the 19th century, as was the fact that if f is piecewise continuous then the series converges at each point of continuity. Perhaps surprisingly, it was not shown until the 1960s that if f is quadratically integrable then the series converges for every value of x except those in some set of measure zero.