A Fourier series is an expression of the form

where this means

Obviously such a sequence of partial sums may or may not converge at a particular value of x.
These series have been important in applied math since the time of Fourier who was an officer in Napoleon’s army. He was interested in studying the flow of heat in cannons and invented the concept to aid him in his study. Since that time, Fourier series and the mathematical problems related to their convergence have motivated the development of modern methods in analysis. As recently as the mid 1960’s a problem related to convergence of Fourier series was solved for the first time and the solution of this problem was a big surprise.^{1} This chapter is on the classical theory of convergence of Fourier series.
If you can approximate a function f with an expression of the form

then the function must have the property f
As just explained, Fourier series are useful for representing periodic functions and no other kind of function.There is no loss of generality in studying only functions which are periodic of period 2π. Indeed, if f is a function which has period T, you can study this function in terms of the function g
 (11.1) 
where
 (11.2) 
Also define the n^{th} partial sum of the Fourier series of f by
 (11.3) 
It may be interesting to see where this formula came from. Suppose then that

multiply both sides by e^{−imx} and take the integral ∫ _{−π}^{π}, so that

Now switch the sum and the integral on the right side even though there is absolutely no reason to believe this makes any sense. Then
In case f is real valued, c_{k} = c_{−k} and so

Letting c_{k} ≡ α_{k} + iβ_{k}

where

which shows that

Therefore, letting a_{k} = 2α_{k} and b_{k} = −2β_{k},

and
 (11.4) 
This is often the way Fourier series are presented in elementary courses where it is only real functions which are to be approximated. However it is easier to stick with Definition 11.2.2.
The partial sums of a Fourier series can be written in a particularly simple form which is presented next.
The function

is called the Dirichlet Kernel
Theorem 11.2.3 The function D_{n} satisfies the following:
Proof:Part 1 is obvious because

Therefore,
Here is a picture of the Dirichlet kernels for n=1,2, and 3
Note they are not nonnegative but there is a large central positive bump which gets larger as n gets larger.
It is not reasonable to expect a Fourier series to converge to the function at every point. To see this, change the value of the function at a single point in