Czechoslovak Mathematical Journal
Daniyal M. IsralovApproximation by
p
FaberLaurent rational functions in the weighted Lebesguespaces
Czechoslovak Mathematical Journal
, Vol. 54 (2004), No. 3, 751765Persistent URL:
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Czechoslovak Mathematical Journal, 54 (129) (2004), 751–765
APPROXIMATION BY
p
FABERLAURENT RATIONAL FUNCTIONSIN THE WEIGHTED LEBESGUE SPACES
✂✁✄☎✝✆✞✁✠✟☛✡☞✌✍✏✎✁✑☎✒✟✓✔
, Balikesir
(
Received December 18, 2001)
Abstract.
Let
L
⊂
C
be a regular Jordan curve. In this work, the approximationproperties of the
p
FaberLaurent rational series expansions in the
ω
weighted Lebesguespaces
L
p
(
L,ω
) are studied. Under some restrictive conditions upon the weight functionsthe degree of this approximation by a
k
th integral modulus of continuity in
L
p
(
L,ω
) spacesis estimated.
Keywords
: Faber polynomial, Faber series, weighted Lebesgue space, weighted Smirnovspace,
k
th modulus of continuity
MSC 2000
: 41A10, 41A25, 41A58, 41A30, 30E10
1. Introduction
Let
L
be a rectiﬁable Jordan curve in the complex plane
C
,
G
:= int
L
and
G
−
:=ext
L
. Without loss of generality we assume that
0
∈
G
. Let also
U
:=
{
w
:

w

<
1
}
,
T
:=
∂U
,
U
−
:=
{
w

:

w

>
1
}
, and let
ϕ
and
ϕ
1
be the conformal mappings of
G
−
and
G
onto
U
−
respectively, normalized by
ϕ
(
∞
) =
∞
,
lim
z
→∞
ϕ
(
z
)
/z >
0
and
ϕ
1
(0) =
∞
,
lim
z
→
0
zϕ
1
(
z
)
>
0
.
The inverse mappings of
ϕ
and
ϕ
1
will be denoted by
ψ
and
ψ
1
, respectively.Later on we assume that
p
∈
(1
,
∞
)
, and denote by
L
p
(
L
)
and
E
p
(
G
)
the set of all measurable complex valued functions such that

f

p
is Lebesgue integrable withrespect to arclength, and the Smirnov class of analytic functions in
G
, respectively.751
Each function
f
∈
E
p
(
G
)
has a nontangential limit almost everywhere (a.e.) on
L
,and if we use the same notation for the nontangential limit of
f
, then
f
∈
L
p
(
L
)
.For
p >
1
,
L
p
(
L
)
and
E
p
(
G
)
are Banach spaces with respect to the norm
f
E
p
(
G
)
=
f
L
p
(
L
)
:=
L

f
(
z
)

p

d
z

1
p
.
For the further properties, see [5, pp. 168–185] and [8, pp. 438–453].The order of polynomial approximation in
E
p
(
G
)
,
p
1
has been studied byseveral authors. In [17], Walsh and Russel gave results when
L
is an analytic curve.For domains with suﬃciently smooth boundary, namely when
L
is a smooth Jordancurve and
θ
(
s
)
, the angle between the tangent and the positive real axis expressed as afunction of arclength
s
, has modulus of continuity
Ω(
θ,s
)
satisfying the Dinismoothcondition(1)
δ
0
Ω(
θ,s
)
s
d
s <
∞
, δ >
0
,
this problem, for
p >
1
, was studied by S.Y. Alper [1].These results were later extended to domains with regular boundary which wedeﬁne in Section 2, for
p >
1
by V.M. Kokilashvili [13], and for
p
1
by J.E. Andersson [2]. Similar problems were also investigated in [10]. Let us emphasize that inthese works, the Faber operator, Faber polynomials and
p
Faber polynomials werecommonly used and the degree of polynomial approximation in
E
p
(
G
)
has been studied by applying various methods of summation to the Faber series of functions in
E
p
(
G
)
. More extensive knowledge about them can be found in [7, pp. 40–57] and[16, pp. 52–236].In [11], for domains with a regularboundary we have constructed the approximantsdirectly as the
n
th partial sums of
p
Faber polynomial series of
f
∈
E
p
(
G
)
, and laterapplying the same method in [3], we have investigated the approximation propertiesof the
n
th partial sums of
p
FaberLaurent rational series expansions in the Lebesguespaces
L
p
(
L
)
. The approximation properties of the
p
Faber series expansions in the
ω
weighted Smirnov class
E
p
(
G,ω
)
of analytic functions in
G
whose boundary is aregular Jordan curve are studied in [12].In this work, when
L
is a regular Jordan curve, the approximation properties of the
p
FaberLaurent rational series expansions in the
ω
weighted Lebesgue spaces
L
p
(
L, ω
)
are studied. Under some restrictive conditions upon weight functions thedegree of this approximation is estimated by a
k
th (
k
1
) integral modulus of continuity in
L
p
(
L,ω
)
spaces. The results to be obtained in this work are also newin the nonweighted case
ω
= 1
.752
We shall denote by
c
constants (in general, diﬀerent in diﬀerent relations) depending only on numbers that are not important for the questions of our interest.
2. New results
For the formulation of new results in detail it is necessary to introduce somedeﬁnitions and auxiliary results.
Deﬁnition 1.
L
is called regular if there exists a number
c >
0
such that forevery
r >
0
,
sup
{
L
∩
D
(
z,r
)

:
z
∈
L
}
cr
, where
D
(
z,r
)
is an open disk withradius
r
and centered at
z
and

L
∩
D
(
z,r
)

is the length of the set
L
∩
D
(
z,r
)
.We denote by
S
the set of all regular Jordan curves in the complex plane.
Deﬁnition 2.
Let
ω
be a weight function on
L
.
ω
is said to satisfy the Muckenhoupt
A
p
condition on
L
if
sup
z
∈
L
sup
r>
0
1
r
L
∩
D
(
z,r
)
ω
(
ζ
)

d
ζ

1
r
L
∩
D
(
z,r
)
[
ω
(
ζ
)]
−
1
/p
−
1

d
ζ

p
−
1
<
∞
.
Let us denote by
A
p
(
L
)
the set of all weight functions satisfying the Muckenhoupt
A
p
condition on
L
.For a weight function
ω
given on
L
we also deﬁne the following function spaces.
Deﬁnition 3.
The set
L
p
(
L,ω
) :=
{
f
∈
L
1
(
L
):

f

p
ω
∈
L
1
(
L
)
}
is called the
ω
weighted
L
p
space.
Deﬁnition 4.
The set
E
p
(
G,ω
) :=
{
f
∈
E
1
(
G
):
f
∈
L
p
(
L,ω
)
}
is called the
ω
weighted Smirnov space of order
p
of analytic functions in
G
.Let
g
∈
L
p
(
T,ω
)
and
ω
∈
A
p
(
T
)
. Since
L
p
(
T,ω
)
is noninvariantwith respect to theusual shift, we consider the following mean value function as a shift for
g
∈
L
p
(
T,ω
)
:
σ
h
g
(
w
) := 12
h
h
−
h
g
(
w
e
it
)d
t,
0
< h <
π
, w
∈
T.
As follows from the continuity of the HardyLittlewood maximal operator in weighted
L
p
(
T,ω
)
spaces, the operator
σ
h
is bounded in
L
p
(
T,ω
)
if
ω
∈
A
p
(
T
)
and thefollowing inequality holds:
σ
h
g
L
p
(
T,ω
)
c
(
p
)
g
L
p
(
T,ω
)
,
1
< p <
∞
.
The last relation is equivalent [15] to the property
lim
h
→
0
σ
h
g
−
g
L
p
(
T,ω
)
= 0
.
Starting from the last two relations we can give the following deﬁnition.753
Deﬁnition 5.
If
g
∈
L
p
(
T,ω
)
and
ω
∈
A
p
(
T
)
, then the function
Ω
p,ω,k
(
g,
·
):[0
,
∞
]
→
[0
,
∞
)
deﬁned by
Ω
p,ω,k
(
g,δ
) := sup
0
<h
i
δi
=1
,
2
,...,k
k
i
=1
(
E
−
σ
h
i
)
g
L
p
(
T,ω
)
,
1
< p <
∞
,
is called the
k
th integral modulus of continuity in the
L
p
(
T,ω
)
space for
g
. Here
E
is the identity operator.Note that the idea of deﬁning such a modulus of continuity srcinates from [18].In [9] this idea was used for investigations of the approximation problems in
L
p
([0
,
2
π
]
,ω
)
spaces. Recently, in [12], to obtain direct theorems of the approximation theory in the weighted Smirnov spaces
E
p
(
G,ω
)
, we have used the same ideafor the case
k
= 1
.It can be shown easily that
Ω
p,ω,k
(
g,
·
)
is a continuous, nonnegative and nondecreasing function satisfying the conditions(2)
lim
δ
→
0
Ω
p,ω,k
(
g,δ
) = 0
,
Ω
p,ω,k
(
g
1
+
g
2
,
·
)
Ω
p,ω,k
(
g
1
,
·
) + Ω
p,ω
(
g
2
,
·
)
.
For an arbitrary function
f
∈
L
p
(
L,ω
)
and a weight function given on
L
we alsoset
f
0
(
w
) :=
f
[
ψ
(
w
)](
ψ
′
(
w
))
1
/p
, f
1
(
w
) :=
f
[
ψ
1
(
w
)](
ψ
′
1
(
w
))
1
/p
w
2
/p
,
(3)
ω
0
(
w
) :=
ω
[
ψ
(
w
)]
, ω
1
(
w
) :=
ω
[
ψ
1
(
w
)]
.
The condition
f
∈
L
p
(
L,ω
)
, implies that
f
0
∈
L
p
(
T,ω
0
)
and
f
1
∈
L
p
(
T,ω
1
)
.Then if
ω
∈
A
p
(
L
)
and
ω
0
,ω
1
∈
A
p
(
T
)
we can deﬁne the weighted integral moduliof continuity
Ω
p,ω,k
(
f
0
,δ
)
and
Ω
p,ω,k
(
f
1
,δ
)
, using the procedure given above.Main result in our work is the following theorem.
Theorem 1.
Let
L
∈
S
and
f
∈
L
p
(
L,ω
)
,
1
< p <
∞
. If
ω
∈
A
p
(
L
)
and
ω
0
,ω
1
∈
A
p
(
T
)
, then for every natural number
n
there are a constant
c >
0
and a rational function
R
n
(
z,f
) :=
n
k
=
−
n
a
(
n
)
k
z
k
such that
f
−
R
n
(
·
,f
)
L
p
(
L,ω
)
c
Ω
p,ω
0
,k
f
0
,
1
n
+ Ω
p,ω
1
,k
f
1
,
1
n
,
754