Advanced Calculus Single Variable

11.2 Definition And Basic Properties

A Fourier series is an expression of the form

∑∞ ikx cke k=−∞

where this means

∑n lim ckeikx. n→∞ k=−n

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

∑∞ ikx cke k=−∞

then the function must have the property f

= f because this is true of every term in the above series. More generally, here is a definition.

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

≡ f where g is periodic of period 2π.

It may be interesting to see where this formula came from. Suppose then that

∞∑ f (x) = ckeikx, k=−∞

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

∫ π ∫ π ∞∑ f (x) e− imxdx = ckeikxe−imxdx. −π −π k=−∞

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

because ∫ _−π^πe^ikxe^−imxdx = 0 if k≠m. It is formal manipulations of the sort just presented which suggest that Definition 11.2.2 might be interesting.

In case f is real valued, c_k = c_−k and so

∫ 1-- π n∑ ( ikx) Snf (x) = 2π − πf (y)dy + 2Re cke . k=1

Letting c_k ≡ α_k + iβ_k

∫ n 1-- π ∑ Snf (x) = 2π − πf (y)dy + 2[αk coskx− βk sinkx] k=1

where

∫ π ∫ π ck = 1-- f (y)e−ikydy =-1 f (y)(cos ky− isin ky)dy 2π − π 2π −π

which shows that

∫ π ∫ π αk = -1- f (y)cos(ky) dy,βk = −-1 f (y)sin (ky)dy 2π −π 2π −π

Therefore, letting a_k = 2α_k and b_k = −2β_k,

∫ π ∫ π ak =-1 f (y)cos(ky)dy,bk = 1 f (y)sin(ky)dy π −π π −π

and

a0 ∑n Snf (x) = 2 + akcoskx + bk sinkx (11.4) k=1

(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

n Dn (t) ≡ 1--∑ eikt 2πk=− n

is called the Dirichlet Kernel

Proof:Part 1 is obvious because

∫ _−π^πe^−ikydy = 0 whenever k≠0 and it equals 1 if k = 0. Part 2 is also obvious because t → e^ikt is periodic of period 2π. It remains to verify Part 3. Note

∑n ∑n 2πDn (t) = eikt = 1+ 2 cos(kt) k=−n k=1

Therefore,

where the easily verified trig. identity cossin = is used to get to the second line. This proves 3 and proves the theorem.

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

and extend to keep the modified function periodic. Then the Fourier series of the modified function is the same as the Fourier series of the original function and so if pointwise convergence did take place, it no longer does. However, it is possible to prove an interesting theorem about pointwise convergence of Fourier series. This is done next.