33.5. POINTWISE CONVERGENCE OF FOURIER SERIES 649

and so, it is obvious that ∫ L

−LDn (t)dt = 1. (33.10)

From the above formula, it follows that Dn (t) = Dn (−t) and Dn (x+2L) = Dn (x). SinceDn (t) = Dn (−t) , ∫ L

−LDn (t)dt = 2

∫ L

0Dn (t)dt

It remains to find a formula. Use 33.2

sin(

π

2Lt)

Dn (t) =1L

(12

sin(

π

2Lt)+

n

∑k=1

sin(

π

2Lt)

cos(

Lt))

=1L

(12

sin(

π

2Lt)+

12

n

∑k=1

sin((

L+

π

2L

)t)− sin

((kπ

L− π

2L

)t))

=1

2L

[sin(

π

2Lt)+

n

∑k=1

sin((

k+12

Lt)−

n

∑k=1

sin((

k− 12

Lt)]

=1

2L

[sin(

π

2Lt)+

n

∑k=1

sin((

k+12

Lt)−

n−1

∑k=0

sin((

k+12

Lt)]

=1

2Lsin((

n+12

Lt)

Thus the desired formula is

Dn (t) =sin((

n+ 12

L t)

2Lsin(

π

2L t) ■

Here is a graph of the first seven of these Dirichlet kernels, n ≥ 1 for L = π .

-4 -2 0 2 4-1

0

1

2Next, it follows from the above that

Sn f (x) =∫ L

−LDn (x− y) f (y)dy.

Change the variables. Let u = x− y. Then this re-duces to ∫ L+x

−L+xDn (u) f (x−u)du

Since Dn and f are both periodic of period 2L, this equals∫ L

−LDn (y) f (x− y)dy

Therefore, since∫ L−L Dn (y)dy = 1,∣∣∣∣ f (x+)+ f (x−)

2−Sn f (x)

∣∣∣∣= ∣∣∣∣ f (x+)+ f (x−)2

−∫ L

−LDn (y) f (x− y)dy

∣∣∣∣

33.5. POINTWISE CONVERGENCE OF FOURIER SERIES 649and so, it is obvious that ji/ D, (t)dt = 1. (33.10)J-LFrom the above formula, it follows that D, (t) = D, (—t) and D, (x+2L) = D, (x). SinceDy (t) = Dn(—t),L “L/ D, (t)dt = 2 | D, (t) dtJ-L 0It remains to find a formula. Use 33.2sin (=) D,(t) = ; (; sin (<1) + y sin (=) cos ())k=1Thus the desired formula is_ sin((n+ 5) Ft)Pal) = SE sin(Bi)Here is a graph of the first seven of these Dirichlet kernels, n > 1 for L = 7.5 Next, it follows from the above thatL; Sif) = [/ Duley) Fay.0 Change the variables. Let u = x — y. Then this re-° ° duces toL+x| / Dy (u) f (x—u)du-4 -2 0 2 4 —E+xSince D, and f are both periodic of period 2Z, this equalsL[> (y) f («—y) dyTherefore, since [“, Dn (y)dy = 1,FONE 5,00) = eer) Daly) fe=y)dy—L