The Real And Complex Number Systems

n
∑
k1
sin kx
k
≤ 1
n 1 ∑ n
k1
≤ 1
n 1
sin kx
k
n
∑
k1
1
sin 2 1
sin 2 1 − 1
n 1
1
sin .
2
(b) 0 x ≤ :LetN 1 x , consider two cases as follows.
As n N, then
and as n ≥ N, then
≤
n
∑
k1
N−1
∑
k1
sin kx
k
sin kx
k
n
≤ 1 ∑
kN
sin kx
k
≤ 1 1
n 1 ∑ n
k1
n
∑
kN
by (*)
sin kx
k
by summation by parts
≤ 1 1
n 1 sin x 2
1 2 .
1 x sin x 2
2
Note that lim x→0
1 x sin x 2
n
∑
k1
sin kx
k
1
N sin x 2
sin kx
k
N−1
1
N ∑ k1
k
∑ sin jx 1
k 1 − 1 k
j1
≤ n|x| N|x| ≤ 1 *
sin kx
k
1
N
− 1
n 1
n
∑
kN
1
sin x 2
4. So, we may choose a ′ such that
k
∑ sin jx 1
k 1 − 1 k
j1
2 ≤ 5 for all x ∈ 0,
1 x sin x ′ .
2
By preceding sayings, we have proved that F n x is uniformly bounded on I. It means
that F n x is uniformly bounded on R.
D In 1911, Otto Toeplitz proves the following. Let a n and x n be two sequences
1
such that a n 0 for all n with lim n→ a 1 ...a n
0 and lim n→ x n x. Then
lim
a 1 x 1 ...a n x n
n→ a 1 ...a n
x.
n
Proof: LetS n ∑ k1
n
a k and T n ∑ k1
a k x k , then
lim
T n1 − T n
n→ S n1 − S n
lim n→
a n1 x n1
a n1
lim n→
x n1 x.
So, by O-Stolz’s Theorem, wehaveproveit.
Remark: (1) Let a n 1, then it is an extension of Theorem 8.48.
(2) Show that