278 CHAPTER 12. SERIES AND TRANSFORMS

It may be interesting to see where this formula came from. Suppose then that f (x) =∑

∞k=−∞

ckeikx, multiply both sides by e−imx and take the integral∫

π

−π, so that∫

π

−π

f (x)e−imxdx =∫

π

−π

∞

∑k=−∞

ckeikxe−imxdx.

Now switch the sum and the integral on the right side even though there is absolutely no rea-son to believe this makes any sense. Then

∫π

−πf (x)e−imxdx = ∑

∞k=−∞

ck∫

π

−πeikxe−imxdx =

cm∫

π

−π1dx = 2πcm because

∫π

−πeikxe−imxdx = 0 if k ̸= m. It is formal manipulations of the

sort just presented which suggest that Definition 12.1.2 might be interesting.In case f is real valued, ck = c−k and so

Sn f (x) =1

2π

∫π

−π

f (y)dy+n

∑k=1

2Re(

ckeikx).

Letting ck ≡ αk + iβ k

Sn f (x) =1

2π

∫π

−π

f (y)dy+n

∑k=1

2 [αk coskx−β k sinkx]

where ck =1

2π

∫π

−πf (y)e−ikydy = 1

2π

∫π

−πf (y)(cosky− isinky)dy which shows that

αk =1

2π

∫π

−π

f (y)cos(ky)dy, β k =−12π

∫π

−π

f (y)sin(ky)dy

Therefore, letting ak = 2αk and bk =−2β k,

ak =1π

∫π

−π

f (y)cos(ky)dy, bk =1π

∫π

−π

f (y)sin(ky)dy

and

Sn f (x) =a0

2+

n

∑k=1

ak coskx+bk sinkx (12.4)

This is often the way Fourier series are presented in elementary courses where it is onlyreal functions which are to be approximated. However it is easier to stick with Definition12.1.2.

The partial sums of a Fourier series can be written in a particularly simple form whichis presented next.

Sn f (x) =n

∑k=−n

ckeikx =n

∑k=−n

(1

2π

∫π

−π

f (y)e−ikydy)

eikx

=∫

π

−π

Dn(x−y)︷ ︸︸ ︷1

2π

n

∑k=−n

(eik(x−y)

)f (y)dy≡

∫π

−π

Dn (x− y) f (y)dy.

The function Dn (t)≡ 12π ∑

nk=−n eikt is called the Dirichlet Kernel.

Theorem 12.1.3 The function Dn satisfies the following: