3.3 THEORY OF FUNCTIONS
237
where the convention ao := 0 and its consequence Da0 = -a1 are used. It follows that the series (2) is convergent: 00
00
E ak sin kx = E Dk (x) Dak =: f (X) k=1
(7)
k=0
for every x with the possible exception of the case x = 0 (mod 27r). In the following, we make use of an inequality of the Sidon type: for any integer n > 2 and numerical sequence {bk}, it
1/2
2n-1
2n-1
E bkDk(x) dx < C > kbk
ir/n
k=n
(8)
k=n
where C is a positive constant. To see this, we first apply the CauchySchwarz inequality and then exploit the orthogonality of the system {cos (k +
2)
x}:
2n-1
it
rir
2n-1
E bkDk(x) dx=1/n k=n 1/2
dx
T'n
2)2 1/2
2
bkcos l k+ 2 l x k=n
<C
dx
\\\
1/2
2n-1
k=n
cos (k + 2) x dx 2 sin 2
2n-1
J0
Cn- > bk
<
k=n
it
(2 sin
(2n_1
bk
kbk
k=n
Let s > 1 be an integer. According to (7),
f
if
j-1 > Dk (x)Dak dx
2'-1
(x) dx >
a2
j=1
J'r/(7+1)
2'-1
k=0
00
E Dk(x)Dak dx := I1 - I2 . 7=1
3T k=7
Using the inequality
<k+1 (0 <x <ir; k=0,1,...), we obtain that 2'-1
I1 >
it/J
9-1
l(7+1) k=0 = I11 - 112 .
dx
Dak X- -
2'-1
1
9=1
ir/7
7-1
> (k + 1) jakl dx
fit/(j+1) k=0
(9)