Optimal information for approximating periodic analytic functions
Authors: Osipenko K. , Wilderotter K. 
Published: July 2002

Let

S

β

:=
{
z
∈

C

:

|

Im
⁡
z

|

>
β
}

S_{\beta }:=\{z\in \mathbb {C}:| \operatorname {Im}z|>\beta \}

be a strip in the complex plane. For fixed integer

r
≥
0

r\ge 0

let

H

∞
,
β

r

H^r_{\infty ,\beta }

denote the class of

2
π

2\pi

-periodic functions

f
f

, which are analytic in

S

β

S_{\beta }

and satisfy

|

f

(
r
)

(
z
)

|

≤
1

|f^{(r)}(z)|\le 1

in

S

β

S_{\beta }

. Denote by

H

∞
,
β

r
,

R

H^{r,\mathbb R}_{\infty ,\beta }

the subset of functions from

H

∞
,
β

r

H^r_{\infty ,\beta }

that are real-valued on the real axis. Given a function

f
∈

H

∞
,
β

r

f\in H^r_{\infty ,\beta }

, we try to recover

f
(
ζ
)

f(\zeta )

at a fixed point

ζ
∈

R

\zeta \in \mathbb {R}

by an algorithm

A
A

on the basis of the information
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I f equals left-parenthesis a 0 left-parenthesis f right-parenthesis comma a 1 left-parenthesis f right-parenthesis comma ellipsis comma a Subscript n minus 1 Baseline left-parenthesis f right-parenthesis comma b 1 left-parenthesis f right-parenthesis comma ellipsis comma b Subscript n minus 1 Baseline left-parenthesis f right-parenthesis right-parenthesis comma">

I
f
=
(

a

0

(
f
)
,

a

1

(
f
)
,
…
,

a

n
−
1

(
f
)
,

b

1

(
f
)
,
…
,

b

n
−
1

(
f
)
)
,

\begin{equation*}If=(a_{0}(f),a_{1}(f),\dots ,a_{n-1}(f),b_{1}(f),\dots ,b_{n-1}(f)), \end{equation*}

where

a

j

(
f
)

a_{j}(f)

,

b

j

(
f
)

b_{j}(f)

are the Fourier coefficients of

f
f

. We find the intrinsic error of recovery
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper H Subscript normal infinity comma beta Superscript r Baseline comma upper I right-parenthesis colon equals inf Underscript upper A colon double-struck upper C Superscript 2 n minus 1 Baseline right-arrow double-struck upper C Endscripts sup Underscript f element-of upper H Subscript normal infinity comma beta Superscript r Baseline Endscripts StartAbsoluteValue f left-parenthesis zeta right-parenthesis minus upper A left-parenthesis upper I f right-parenthesis EndAbsoluteValue period">

E
(

H

∞
,
β

r

,
I
)
:=

inf

A
:

C

2
n
−
1

→

C

sup

f
∈

H

∞
,
β

r

|

f
(
ζ
)
−
A
(
I
f
)

|

.

\begin{equation*} E(H^r_{\infty ,\beta } ,I):= \operatorname * {inf}_{A\colon \mathbb {C}^{2n-1}\to \mathbb {C}}\,\sup _{f\in H^r_{\infty ,\beta } }|f(\zeta )- A(If)|. \end{equation*}

Furthermore the

(
2
n
−
1
)

(2n-1)

-dimensional optimal information error, optimal sampling error and

n
n

-widths of

H

∞
,
β

r
,

R

H^{r,\mathbb R}_{\infty ,\beta }

in

C
C

, the space of continuous functions on

[
0
,
2
π
]

[0,2\pi ]

, are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class

H

p
,
β

H_{p,\beta }

, consisting of all

2
π

2\pi

-periodic functions, which are analytic in

S

β

S_{\beta }

with

p
p

-integrable boundary values. In the case

p
=
2

p=2

sampling fails to yield optimal information as well in odd as in even dimensions.