Let
S
β
:=
{
z
∈
C
:
|
Im
z
|
>
β
}
S_{\beta }:=\{z\in {\mathbb C}:|\textrm {Im}z|>\beta \}
be a strip in complex plane.
H
~
∞
,
β
r
\widetilde {H}_{\infty ,\beta }^{r}
denotes those
2
π
2\pi
-periodic, real-valued functions on
R
{\mathbb R}
which are analytic in the strip
S
β
S_{\beta }
and satisfy the condition
|
f
(
r
)
(
z
)
|
≤
1
|f^{(r)}(z)|\leq 1
,
z
∈
S
β
z\in S_{\beta }
. Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel′fand, and information
n
n
-widths of
H
~
∞
,
β
r
\widetilde {H}_{\infty ,\beta }^{r}
in
L
∞
[
0
,
2
π
]
L_{\infty }[0,2\pi ]
,
r
=
0
,
1
,
2
,
…
r=0,1,2,\ldots
, and 2
n
n
-widths of
H
~
∞
,
β
r
\widetilde {H}_{\infty ,\beta }^{r}
in
L
q
[
0
,
2
π
]
L_{q}[0,2\pi ]
,
r
=
0
r=0
,
1
≤
q
>
∞
1\leq q>\infty
. In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on
H
~
∞
,
β
r
\widetilde {H}_{\infty ,\beta }^{r}
, from which we get an inequality of Landau–Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel′fand
n
n
-width of
H
~
∞
,
β
r
\widetilde {H}_{\infty ,\beta }^{r}
in
L
q
[
0
,
2
π
]
L_{q}[0,2\pi ]
,
r
=
0
,
1
,
2
…
,
r=0,1,2\ldots ,
1
≤
q
>
∞
1\leq q>\infty
. Finally, we calculate the exact values of Kolmogorov
2
n
2n
-width, linear
2
n
2n
-width, and information
2
n
2n
-width of
H
~
∞
,
β
r
\widetilde {H}_{\infty ,\beta }^{r}
in
L
q
[
0
,
2
π
]
L_{q}[0,2\pi ]
,
r
∈
N
r\in {\mathbb N}
,
1
≤
q
>
∞
1\leq q>\infty
.