12.2. CRITERIA FOR CONVERGENCE 279
1.∫
π
−πDn (t)dt = 1
2. Dn is periodic of period 2π
3. Dn (t) = (2π)−1 sin(n+ 12 )t
sin( t2 )
.
Proof: Part 1 is obvious because 12π
∫π
−πe−ikydy = 0 whenever k ̸= 0 and it equals 1 if
k = 0. Part 2 is also obvious because t→ eikt is periodic of period 2π since
eik(t+2π) = cos(kt +2πk)+ isin(kt +2πk) = cos(kt)+ isin(kt) = eikt
It remains to verify Part 3. Note 2πDn (t) = ∑nk=−n eikt = 1+2∑
nk=1 cos(kt) . Therefore,
2πDn (t)sin( t
2
)= sin
( t2
)+2
n
∑k=1
sin( t
2
)cos(kt)
= sin( t
2
)+
n
∑k=1
sin((
k+12
)t)− sin
((k− 1
2
)t)= sin
((n+
12
)t)
where the easily verified trig. identity cos(a)sin(b) = 12 (sin(a+b)− sin(a−b)) 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 = 1,2,3 and 4
-4 -2 0 2 4
0
0.5
1
1.5
Note they are not nonnegative but there is a large central positive bump which getslarger 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 (−π,π) and extend to keepthe modified function periodic. Then the Fourier series of the modified function is the sameas 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 pointwiseconvergence of Fourier series. This is done next.
12.2 Criteria for ConvergenceFourier series like to converge to the midpoint of the jump of a function under suitableconditions. This was first shown by Dirichlet in 1829 after others had tried unsuccessfullyto prove such a result. The condition given for convergence in the following theorem isdue to Dini. [3] It is a generalization of the usual theorem presented in elementary bookson Fourier series methods. Fourier did not appreciate the difficulty of this question andwas happy to believe that the series did converge to the function in some useable sensedespite the doubts of people like Lagrange and Laplace. For over 150 years they studiedthis question and major results appeared as recently as the mid 1960’s.