Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes
UDC 517.5 Let $G\subset \mathbb{C}$ be a doubly connected domain bounded by two rectifiable Carleson curves. In this work, we use the higher modulus of smoothness in order to investigate the approximation properties of $(p-\varepsilon)$-Faber–Laurent rational functions in the subclass o...
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| author | Testici, A. Testici, Ahmet Testici, A. |
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Let $G\subset \mathbb{C}$ be a doubly connected domain bounded by two rectifiable Carleson curves. In this work, we use the higher modulus of smoothness in order to investigate the approximation properties of $(p-\varepsilon)$-Faber–Laurent rational functions in the subclass of weighted generalized grand Smirnov classes ${E}^{p),\theta } ( {G,\omega })$ of analytic functions. |
| doi_str_mv | 10.37863/umzh.v73i7.559 |
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DOI: 10.37863/umzh.v73i7.559
UDC 517.5
A. Testici (Balikesir Univ., Turkey)
APPROXIMATION BY RATIONAL FUNCTIONS
ON DOUBLY CONNECTED DOMAINS
IN WEIGHTED GENERALIZED GRAND SMIRNOV CLASSES
НАБЛИЖЕННЯ РАЦIОНАЛЬНИМИ ФУНКЦIЯМИ
ДЛЯ ДВОЗВ’ЯЗНИХ ОБЛАСТЕЙ У ЗВАЖЕНИХ
УЗАГАЛЬНЕНИХ ВЕЛИКИХ КЛАСАХ СМIРНОВА
Let G \subset \BbbC be a doubly connected domain bounded by two rectifiable Carleson curves. In this work, we use the higher
modulus of smoothness in order to investigate the approximation properties of (p - \varepsilon )-Faber – Laurent rational functions
in the subclass of weighted generalized grand Smirnov classes Ep),\theta (G,\omega ) of analytic functions.
Нехай G \subset \BbbC — двозв’язна область, що обмежена двома спрямлюваними кривими Карлесона. У цiй роботi за
допомогою вищого модуля гладкостi вивчаються апроксимацiйнi властивостi рацiональних (p - \varepsilon )-функцiй Фабера –
Лорана у пiдкласах зважених узагальнених великих класiв Смiрнова Ep),\theta (G,\omega ) аналiтичних функцiй.
1. Introduction. We assume that B is a simply connected domain, bounded by a rectifiable Jordan
curve \Gamma . We denote by Lp(\Gamma ) and Ep(B), 1 \leq p < \infty , the set of all measurable complex valued
functions such that | f | p is Lebesgue integrable with respect to arclength on \Gamma , and the Smirnov class
of analytic functions in B, respectively. We recall that if there exists a sequence (\gamma n), n = 1, 2, . . . ,
of rectifiable Jordan curves in B, which converges to \Gamma as n\rightarrow \infty such that
\mathrm{s}\mathrm{u}\mathrm{p}
n
\left\{
\int
\gamma n
\bigm| \bigm| f(z)\bigm| \bigm| p| dz|
\right\} <\infty ,
we say that f belongs to Smirnov class Ep(B) [24, p. 168]. Each function f \in Ep(B) has non-
tangential limit almost everywhere (a.e.) on \Gamma and if we use the same notation for the limit function
of f, then f \in Lp(\Gamma ). Lp(\Gamma ) and Ep(B) are Banach spaces with respect to the norm
\| f\| Ep(B) := \| f\| Lp(\Gamma ) :=
\left( \int
\Gamma
\bigm| \bigm| f(z)\bigm| \bigm| p| dz|
\right) 1/p
, 1 \leq p <\infty .
Let G \subset \BbbC be a doubly connected domain in the complex plane \BbbC , bounded by a rectifiable
Jordan curves \Gamma 1 and \Gamma 2 which \Gamma 2 is in \Gamma 1.
Let G -
1 := \mathrm{E}\mathrm{x}\mathrm{t} \Gamma 1, G1 := \mathrm{I}\mathrm{n}\mathrm{t} \Gamma 1 and G -
2 := \mathrm{E}\mathrm{x}\mathrm{t} \Gamma 2, G2 := \mathrm{I}\mathrm{n}\mathrm{t} \Gamma 2. Without loss of generality
we suppose that 0 \in G2.
Let also \BbbT :=
\bigl\{
w \in \BbbC : | w| = 1
\bigr\}
, \BbbU := \mathrm{I}\mathrm{n}\mathrm{t}\BbbT and \BbbU - := \mathrm{E}\mathrm{x}\mathrm{t}\BbbT . We denote by \varphi and \varphi 1 the
conformal mappings of G -
1 and G2 onto \BbbU - , respectively, normalized by
\varphi (\infty ) = \infty , \mathrm{l}\mathrm{i}\mathrm{m}
z\rightarrow \infty
\varphi (z)
z
> 0, and \varphi 1(0) = \infty , \mathrm{l}\mathrm{i}\mathrm{m}
z\rightarrow 0
z\varphi 1(z) > 0.
c\bigcirc A. TESTICI, 2021
964 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 965
Let \psi and \psi 1 be the inverse mappings of \varphi and \varphi 1, respectively. The functions \varphi and \psi
have continuous extensions to \Gamma 1 and \BbbT , their derivatives \varphi \prime and \psi \prime have definite nontangential
limit values a.e. on \Gamma 1 and \BbbT , they are integrable with respect to Lebesgue measure on \Gamma 1 and
\BbbT , respectively. Similarly, the functions \varphi 1 and \psi 1 have continuous extensions to \Gamma 2 and \BbbT , their
derivatives \varphi \prime
1 and \psi \prime
1 have definite nontangential limit values a.e. on \Gamma 2 and \BbbT , which are integrable
with respect to Lebesgue measure on \Gamma 2 and \BbbT [16, p. 19 – 438].
We set
Lr :=
\bigl\{
z \in G -
1 : | \varphi (z)| = r > 1
\bigr\}
and LR :=
\bigl\{
z \in G2 : | \varphi (z)| = R > 1
\bigr\}
.
Let G -
r := \mathrm{E}\mathrm{x}\mathrm{t}Lr, Gr := \mathrm{I}\mathrm{n}\mathrm{t}Lr and G -
R := \mathrm{E}\mathrm{x}\mathrm{t}LR, GR := \mathrm{I}\mathrm{n}\mathrm{t}LR.
\varphi is analytic function in G -
r and \bigl[
\varphi (z)
\bigr] k \bigl[
\varphi \prime (z)
\bigr] 1/(p - \varepsilon )
has a pole with kth degree at \infty , where 0 < \varepsilon < p - 1. Besides \varphi 1 is analytic function in GR and\bigl[
\varphi 1(z)
\bigr] k - 2
p - \varepsilon
\bigl[
\varphi \prime
1(z)
\bigr] 1/(p - \varepsilon )
has a pole with kth degree at 0, where 0 < \varepsilon < p - 1. For construction of polynomials of approxi-
mation process, we need some expansions. For this purpose, applying the same technic used in [5],
for 1 < p <\infty and 0 < \varepsilon < p - 1, we obtain\bigl[
\psi \prime (w)
\bigr] 1 - 1
p - \varepsilon
\psi (w) - z
=
\infty \sum
k=0
Fk,p,\varepsilon (z)
wk+1
, z \in Gr, w \in \BbbU - ,
w
- 2
p - \varepsilon
\bigl[
\psi \prime
1(w)
\bigr] 1 - 1
p - \varepsilon
\psi 1(w) - z
=
\infty \sum
k=0
-
\widetilde Fk,p,\varepsilon (1/z)
wk+1
, z \in G -
R, w \in \BbbU - ,
where Fk,p,\varepsilon (z) and \widetilde Fk,p,\varepsilon (1/z) are polynomials with respect to z and 1/z, respectively. Note
that, firstly, Fk,p,\varepsilon (z) and \widetilde Fk,p,\varepsilon (1/z) were stated in [14]. As in the classical case Fk,p,\varepsilon (z) and\widetilde Fk,p,\varepsilon (1/z) have the following integral representations for every k = 0, 1, 2, . . .:
Fk,p,\varepsilon (z) =
1
2\pi i
\int
Lr
\bigl[
\varphi (\zeta )
\bigr] k\bigl(
\varphi \prime (\zeta )
\bigr) 1
p - \varepsilon
\zeta - z
d\zeta , z \in Gr, r > 1, (1)
\widetilde Fk,p,\varepsilon (1/z) = - 1
2\pi i
\int
LR
\bigl[
\varphi 1(\xi )
\bigr] k - 2/p - \varepsilon
(\varphi \prime
1(\xi ))
1
p - \varepsilon
\xi - z
d\xi , z \in G -
R, R > 1. (2)
The polynomials Fk,p,\varepsilon (z) and \widetilde Fk,p,\varepsilon (1/z) are called the (p - \varepsilon )-Faber polynomials for Gr and G -
R,
respectively.
If f is analytic function in doubly connected domain bounded by curves Lr and LR, then, for
k = 0, 1, 2, . . . , using Cauchy’s integral formulae and the expansions given for Fk,p,\varepsilon and \widetilde Fk,p,\varepsilon , we
have the (p - \varepsilon )-Faber – Laurent series expansion
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
966 A. TESTICI
f(z) =
\infty \sum
k=0
ak (f)Fk,p,\varepsilon (z) +
\infty \sum
k=1
\widetilde ak (f) \widetilde Fk,p,\varepsilon (1/z),
where
ak(f) :=
1
2\pi i
\int
| w| =r1
f
\bigl[
\psi (w)
\bigr] \bigl(
\psi \prime (w)
\bigr) 1/(p - \varepsilon )
wk+1
dw, 1 < r1 < r,
and
\widetilde ak(f) := 1
2\pi i
\int
| w| =R1
f
\bigl[
\psi 1(w)
\bigr] \bigl(
\psi \prime
1(w)
\bigr) 1/(p - \varepsilon )
w2/(p - \varepsilon )
wk+1
dw, 1 < R1 < R.
The rational function
Rn(f)(z) :=
n\sum
k=0
ak(f)Fk,p,\varepsilon (z) +
n\sum
k=1
\widetilde ak(f) \widetilde Fk,p,\varepsilon (1/z)
is called the (p - \varepsilon )-Faber – Laurent rational function of degree n of f.
Definition 1. A rectifiable Jordan curve \Gamma is called Carleson curve if the condition
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \Gamma
\mathrm{s}\mathrm{u}\mathrm{p}
r>0
\bigm| \bigm| \Gamma (z, r)\bigm| \bigm|
r
<\infty
holds, where \Gamma (z, r) is portion of \Gamma in the open disk of radius r centered at z and
\bigm| \bigm| \Gamma (z, r)\bigm| \bigm| is it’s
length. We denote by S the set of all Carleson curves.
Under the various conditions on boundary of simply connected domains the direct and converse
theorems of approximation theory in weighted and non weighted Smirnov classes have been investi-
gated widely. When \Gamma is an analytic curve some results were obtained by Walsh and Russel in [19].
In the case of \Gamma is Dini-smooth curve the direct and inverse theorems were proved by S. Y. Alper
in [27]. In Smirnov classes when \Gamma is Carleson curve these results are generalized in [18] and
in weighted Smirnov classes some similar results for Carleson curves were obtained in [5 – 9, 17].
Similar theorems of approximation theory in Smirnov – Orlicz classes were studied in [26, 30, 31, 34].
When \Gamma is Dini-smooth curve direct and inverse theorems of approximation theory in the Smirnov
classes with variable exponent were proved in [10, 12] and earlier similar results are stated without
proof in [15, 25]. The approximation properties of Faber – Laurent series in Lebesgue space with
variable exponent were investigated in [11].
On doubly connected domain, bounded by two Carleson curve, the rate of approximation by
p-Faber – Laurent rational functions in Smirnov classes was studied in [29]. On doubly connected
domain, bounded by Dini-smooth curves, the rate of approximation by Faber rational functions in
Smirnov – Orlicz classes and Smirnov classes with variable exponent were investigated in [28] and
[3], respectively.
The direct and inverse theorems of approximation theory in the weighted generalized grand
Lebesgue spaces were proved in [13]. After that in weighted generalized grand Smirnov classes,
defined on simple connected domain bounded by Carleson curve, some approximation theorems
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 967
were proved in [14]. In this work, we investigate the approximation property of so-called (p - \varepsilon )-
Faber – Laurent rational functions in the weighted generalized grand Smirnov classes, defined on
doubly connected domains.
The set of all measurable functions f such that
\mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - 1
\left\{ \varepsilon \theta 1
| \Gamma |
\int
\Gamma
| f(x)| p - \varepsilon \omega (x)dx
\right\}
1/(p - \varepsilon )
<\infty
constitute weighted generalized grand Lebesgue space Lp),\theta (\Gamma , \omega ). It becomes a Banach space
equipped with the norm
\| f\| Lp),\theta (\Gamma ,\omega ) := \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - 1
\left\{ \varepsilon \theta 1
| \Gamma |
\int
\Gamma
\bigm| \bigm| f(x)\bigm| \bigm| p - \varepsilon
\omega (x)dx
\right\}
1/(p - \varepsilon )
.
If \theta = 0, then Lp),\theta (\Gamma ) turns into classical Lebesgue space Lp(\Gamma ). In nonweighted case, when
\theta = 1, Lp),\theta (\Gamma ) is called grand Lebesgue space and it is denoted by Lp)(\Gamma ). The spaces Lp),\theta (\Gamma )
were introduced for \theta = 1 in [32] and for \theta > 1 in [21]. Dual spaces of Lp)(\Gamma ) were characterized
in [1] and at the same work was given that Lp)(\Gamma ) is rearrangement invariant and Banach function
space but it is not reflexive. We can show that
Lp(\Gamma ) \subset Lp)(\Gamma ) \subset Lp - \varepsilon (\Gamma ).
We can say that similar embedding relations hold in case of weighted generalized grand Lebesgue
space: if \theta 1 < \theta 2 and 1 < p <\infty , then the embeddings
Lp(\Gamma , \omega ) \subset Lp),\theta 1(\Gamma , \omega ) \subset Lp),\theta 2(\Gamma , \omega ) \subset Lp - \varepsilon (\Gamma , \omega )
are valid.
Lp(\Gamma , \omega ) is not dense in Lp),\theta (\Gamma , \omega ). We denote by \scrL p),\theta (\Gamma , \omega ) the closure of Lp(\Gamma , \omega ) with
respect to the norm of Lp),\theta (\Gamma , \omega ). We state that (see [20, 22]) \scrL p)(\Gamma ) is the set of functions
satisfying the condition
\mathrm{l}\mathrm{i}\mathrm{m}
\varepsilon \rightarrow 0
\left( \varepsilon \theta 1
| \Gamma |
\int
\Gamma
\bigm| \bigm| f(x)\bigm| \bigm| p - \varepsilon
\omega (x)dx
\right) = 0.
Now we construct the Smirnov class defined on the doubly connected domains. Let G\ast be a
doubly connected domain in \BbbC and f be an analytic function in G\ast . If there exists a sequence
(\Delta \nu )
\infty
\nu =1 of domains whose boundaries (\Gamma \nu )
\infty
\nu =1 consist of two rectifiable Jordan curves, lengths
of (\Gamma \nu )
\infty
\nu =1 are bounded, such that the domain \Delta n contains each compact subset of G\ast for every
n \geq N, for some n \in \BbbN and
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
\nu \rightarrow \infty
\left\{
\int
\Gamma \nu
| f(z)| p| dz|
\right\} <\infty ,
then it is said that f belongs to Smirnov classes Ep(G\ast ), p \geq 1 [24, p. 182].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
968 A. TESTICI
Definition 2. Let \Gamma := \Gamma 1 \cup \Gamma -
2 and G be a doubly connected domain bounded by \Gamma 1 and
\Gamma 2 \in S, where \Gamma 2 is in \Gamma 1. Let \omega be a weight function on \Gamma . The set
Ep),\theta (G,\omega ) :=
\Bigl\{
f \in E1 (G) : f \in Lp),\theta (\Gamma , \omega )
\Bigr\}
is called the weighted generalized grand Smirnov class of analytic functions in G.
For f \in Ep),\theta (G,\omega ) norm is defined by \| f\| Ep),\theta (G,\omega ) := \| f\| Lp),\theta (\Gamma ,\omega ). We denote by \scrE p),\theta (G,\omega )
the closure of Smirnov class Ep(G,\omega ) of analytic function with respect to norm Ep),\theta (G,\omega ).
The Cauchy singular integral S\Gamma (f) and the Hardy – Littlewood maximal function M\Gamma (f) for
f \in L1(\Gamma ) for almost all z0 \in \Gamma are defined as following:
S\Gamma (f)(z0) := \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 0
\int
\Gamma \setminus \Gamma (z0,r)
f(z)
z - z0
dz and M\Gamma (f)(z0) := \mathrm{s}\mathrm{u}\mathrm{p}
r>0
1
r
\int
\Gamma (z0,r)
| f(z)| | dz| .
Definition 3. Let \omega be weight function on \Gamma such that \Gamma \in S. Let 1 < p <\infty and 1/p+1/q =
= 1. We say that \omega satisfies Muckenhoupt’s Ap condition on \Gamma if
\mathrm{s}\mathrm{u}\mathrm{p}
z0\in \Gamma
\mathrm{s}\mathrm{u}\mathrm{p}
r>0
\left( 1
r
\int
\Gamma (z0,r)
\omega (z)| dz|
\right)
\left( 1
r
\int
\Gamma (z0,r)
\bigl[
\omega (z)
\bigr] - 1/(p - 1)| dz|
\right)
p - 1
<\infty .
Theorem A [33]. Let \Gamma \in \scrS , 1 < p <\infty and \theta > 0. The operators S\Gamma : f \rightarrow S\Gamma (f) and M\Gamma :
f \rightarrow M\Gamma (f) are bounded in Lp),\theta (\Gamma , \omega ) if and only if \omega \in Ap(\Gamma ).
The norm in Lp),\theta (\BbbT , \omega ) space of 2\pi -periodic functions f is defined as
\| f\| Lp),\theta (\BbbT ,\omega ) := \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - 1
\left\{ \varepsilon \theta
2\pi
2\pi \int
0
\bigm| \bigm| f \bigl( eit\bigr) \bigm| \bigm| p - \varepsilon
\omega
\bigl(
eit
\bigr)
dt
\right\}
1/(p - \varepsilon )
.
Let f \in Lp),\theta (\BbbT , \omega ), 1 < p <\infty , \theta > 0, and, for r = 1, 2, 3, . . . ,
\Delta r
tf(w) =
r\sum
s=0
( - 1)r+s+1
\biggl(
r
s
\biggr)
f
\bigl(
weist
\bigr)
, t > 0.
We define an operator \sigma rhf(w) :=
1
h
\int h
0
\bigm| \bigm| \Delta r
tf(w)
\bigm| \bigm| dt. Let now 0 < h < \infty . For a given \omega \in
\in Ap(\BbbT ), 1 < p < \infty , \theta > 0, by using Theorem A, we get that \mathrm{s}\mathrm{u}\mathrm{p}| h| \leq \delta
\bigm\| \bigm\| \sigma rhf(w)\bigm\| \bigm\| Lp),\theta (\BbbT ,\omega ) \leq
\leq c\| f\| Lp),\theta (\BbbT ,\omega ) <\infty which implies the correctness of the following definition.
Definition 4. Let 1 < p < \infty , \theta > 0 and let f \in Lp),\theta (\BbbT , \omega ), \omega \in Ap(\BbbT ), \delta > 0. The function
\Omega r (f, .)p),\theta ,\omega : [0,\infty ) \rightarrow [0,\infty ) defined by
\Omega r(f, \delta )p),\theta ,\omega := \mathrm{s}\mathrm{u}\mathrm{p}
| h| \leq \delta
\bigm\| \bigm\| \sigma rhf(w)\bigm\| \bigm\| Lp),\theta (\BbbT ,\omega )
is called rth mean modulus of f.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 969
Let \Gamma 1,\Gamma 2 \in \scrS and \omega be a weight function on \Gamma 1 \cup \Gamma -
2 . We can consider \omega as a weight on \Gamma 1
and \Gamma 2, separately.
For any f \in Lp),\theta
\bigl(
\Gamma 1, \omega
\bigr)
and \omega \in Ap(\Gamma 1), we set
f0(w) := f
\bigl[
\psi (w)
\bigr] \bigl(
\psi \prime (w)
\bigr) 1/(p - \varepsilon )
, \omega 0(w) := \omega
\bigl[
\psi (w)
\bigr]
, (3)
and, for any f \in Lp),\theta (\Gamma 2, \omega ) and \omega \in Ap(\Gamma 2), we set
f1(w) := f
\bigl[
\psi 1(w)
\bigr] \bigl(
\psi \prime
1(w)
\bigr) 1
p - \varepsilon w
2
p - \varepsilon , \omega 1(w) := \omega
\bigl[
\psi 1(w)
\bigr]
w - 2. (4)
In this case obviously we have f0 \in Lp),\theta (\BbbT , \omega 0) and f1 \in Lp),\theta (\BbbT , \omega 1) .
Let f \in E1(B), where B is a simply connected domain bounded with the rectifiable Jordan
curve \Gamma \ast . Then f has a nontangential limit a.e. on \Gamma \ast and the boundary function belongs to L1(\Gamma \ast ).
For a given f \in Lp),\theta (\Gamma \ast , \omega ) the functions f+ and f - defined by
f+(z) :=
1
2\pi i
\int
\Gamma \ast
f(\zeta )
\zeta - z
d\zeta =
1
2\pi i
\int
\BbbT
\bigl[
\psi \prime (w)
\bigr] 1 - 1
p - \varepsilon
\psi (w) - z
f0(w)dw, z \in B,
f - (z) :=
1
2\pi i
\int
\Gamma \ast
f(\zeta )
\zeta - z
d\zeta =
1
2\pi i
\int
\BbbT
w
- 2
p - \varepsilon
\bigl[
\psi \prime
1(w)
\bigr] 1 - 1
p - \varepsilon
\psi 1(w) - z
f1(w)dw, z \in B - ,
are analytic in B and B - , respectively, and f - (\infty ) = 0. The functions f+ and f - have the
nontangential limits a.e. on \Gamma and the formulae
f+(z) = S\Gamma (f)(z) +
1
2
f(z) and f - (z) = S\Gamma (f)(z) -
1
2
f(z) (5)
hold. Hence,
f(z) = f+(z) - f - (z) (6)
holds a.e. on \Gamma [16].
The main result of this paper is the following theorem.
Theorem 1. Let \Gamma 1,\Gamma 2 \in \scrS and G be a finite doubly connected domain bounded by \Gamma 1 and \Gamma 2
such that the curve \Gamma 2 is inside of \Gamma 1. Let \Gamma := \Gamma 1\cup \Gamma -
2 and \omega \in Ap(\Gamma ), \omega 0 \in Ap(\BbbT ), \omega 1 \in Ap(\BbbT ),
1 < p < \infty , \theta > 0. If f \in \scrE p),\theta (G,\omega ), then there is a positive constant c independent of n such
that
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma ,\omega )
\leq c
\Biggl[
\Omega r
\biggl(
f0,
1
n
\biggr)
p),\theta ,\omega 0
+\Omega r
\biggl(
f1,
1
n
\biggr)
p),\theta ,\omega 1
\Biggr]
for r = 1, 2, 3, . . . , where Rn(f) is the nth partial sum of the (p - \varepsilon )-Faber – Laurent series of f.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
970 A. TESTICI
2. Auxiliary results. We shall denote by c, c1, . . . , the constants (in general, different in
different relations) depending only on numbers that are not important for the questions of our interest.
The some properties of Faber polynomials were investigated in [2, 4, 23]. Similarly, to p-Faber
polynomials (see [5]) we express some integral representations of Fk,p,\varepsilon (z) and \widetilde Fk,p,\varepsilon (1/z):
If z \in G -
r , then
Fk,p,\varepsilon (z) =
\bigl[
\varphi (z)
\bigr] k\bigl[
\varphi \prime (z)
\bigr] 1/(p - \varepsilon )
+
1
2\pi i
\int
Lr
\bigl[
\varphi (\zeta )
\bigr] k\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
\zeta - z
d\zeta , (7)
and if z \in GR, then
\widetilde Fk,p,\varepsilon
\biggl(
1
z
\biggr)
=
\bigl[
\varphi 1(z)
\bigr] k - 2
p - \varepsilon
\bigl[
\varphi \prime
1(z)
\bigr] 1/(p - \varepsilon ) -
- 1
2\pi i
\int
LR
\bigl[
\varphi 1(\xi )
\bigr] k - 2
p - \varepsilon
\bigl[
\varphi \prime
1(\xi )
\bigr] 1/(p - \varepsilon )
\xi - z
d\xi . (8)
By using Cauchy integral formulae, we have
f(z) =
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z
d\zeta - 1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z
d\xi , z \in G.
If z \in G2 or z \in G -
1 , then
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z
d\zeta - 1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z
d\xi = 0. (9)
We define
I1(z) :=
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z
d\zeta and I2(z) :=
1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z
d\xi .
The function I1 determines the analytic functions I+1 and I - 1 for z \in G1 and z \in G -
1 , respectively,
while the function I2 determines the analytic functions I+2 and I - 2 for z \in G2 and z \in G -
2 ,
respectively.
Lemma 1 [14]. Let \Gamma \in \scrS , \omega \in Ap(\Gamma ), 1 < p < \infty and \theta > 0. If f \in Lp),\theta (\Gamma , \omega ), then
f+ \in Ep),\theta (G,\omega ) and f - \in Ep),\theta (G - , \omega ).
For f0 \in Lp),\theta (\BbbT , \omega ) and \omega 0 \in Ap(\BbbT ), Lemma 1 implies that f+0 \in Ep),\theta (\BbbU , \omega 0) and f - 0 \in
\in Ep),\theta (\BbbU - , \omega 0) such that f - 0 (\infty ) = 0. Similarly, for f1 \in Lp),\theta (\BbbT , \omega ) and \omega 1 \in Ap(\BbbT ), Lemma 1
implies that f+1 \in Ep),\theta (\BbbU , \omega 1) and f - 1 \in Ep),\theta (\BbbU - , \omega 1) such that f - 0 (\infty ) = 0. Then by (6), for
k = 0, 1, 2, 3, . . . , we have
ak(f) =
1
2\pi i
\int
\BbbT
f+0 (w)
wk+1
dw - 1
2\pi i
\int
\BbbT
f - 0 (w)
wk+1
dw =
1
2\pi i
\int
\BbbT
f+0 (w)
wk+1
dw
and
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APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 971
\widetilde ak(f) = 1
2\pi i
\int
\BbbT
f+1 (w)
wk+1
dw - 1
2\pi i
\int
\BbbT
f - 1 (w)
wk+1
dw =
1
2\pi i
\int
\BbbT
f+1 (w)
wk+1
dw.
Hence, ak and \widetilde ak, k = 1, 2, . . . , are Taylor coefficients of f+0 \in Ep),\theta (\BbbU , \omega 0) and f+1 \in Ep),\theta (\BbbU , \omega 1),
respectively.
Lemma 2. Let \omega \in Ap(\BbbT ), 1 < p < \infty , and \theta > 0. If g \in \scrL p),\theta (\BbbT , \omega ), then \Omega r(g
+, \cdot )p),\theta ,\omega \leq
\leq c\Omega r(g, \cdot )p),\theta ,\omega for r = 1, 2, 3, . . . .
Proof. Let g \in \scrL p),\theta (\BbbT , \omega ). Firstly, we show that
\Omega r
\bigl(
S\BbbT (g), \cdot
\bigr)
p),\theta ,\omega
\leq c\Omega r(g, \cdot )p),\theta ,\omega .
By using variate transformation \zeta = ueist and Fubini theorem, we get
\sigma rh
\bigl[
S\BbbT (g)(w)
\bigr]
=
1
h
h\int
0
\Delta r
tS\BbbT (g(w))dt =
=
1
h
h\int
0
r\sum
s=0
( - 1)r+s+1
\biggl(
r
s
\biggr)
S\BbbT (g
\bigl(
weist
\bigr)
dt =
=
1
h
h\int
0
r\sum
s=0
( - 1)r+s+1
\biggl(
r
s
\biggr) \left\{ 1
2\pi i
(P.V )
\int
\BbbT
g(\zeta )
\zeta - weist
d\zeta
\right\} dt =
=
1
h
h\int
0
r\sum
s=0
( - 1)r+s+1
\biggl(
r
s
\biggr) \left\{ 1
2\pi i
(P.V )
\int
\BbbT
g(ueist)
ueist - weist
eistdu
\right\} dt =
=
1
h
h\int
0
r\sum
s=0
( - 1)r+s+1
\biggl(
r
s
\biggr) \left\{ 1
2\pi i
(P.V )
\int
\BbbT
g(ueist)
u - w
du
\right\} dt =
=
1
2\pi i
(P.V )
\int
\BbbT
\biggl\{
1
h
\int h
0
\sum r
s=0
( - 1)r+s+1
\biggl(
r
s
\biggr)
g(ueist)dt
\biggr\}
u - w
du =
=
1
2\pi i
(P.V )
\int
\BbbT
\biggl\{
1
h
\int h
0
\Delta r
t (g(u)dt
\biggr\}
u - w
du = S\BbbT
\bigl[
\sigma rhg(w)
\bigr]
.
Taking norm and supremum over h \leq \delta and applying Theorem A, we have
\Omega r
\bigl(
S\BbbT (g), \cdot
\bigr)
p),\theta ,\omega
= \mathrm{s}\mathrm{u}\mathrm{p}
h\leq \delta
\bigm\| \bigm\| \sigma rh[S\BbbT (g)(w)]\bigm\| \bigm\| Lp),\theta (\BbbT ,\omega ) =
= \mathrm{s}\mathrm{u}\mathrm{p}
h\leq \delta
\bigm\| \bigm\| S\BbbT [\sigma rhg(w)]\bigm\| \bigm\| Lp),\theta (\BbbT ,\omega ) \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
972 A. TESTICI
\leq \mathrm{s}\mathrm{u}\mathrm{p}
h\leq \delta
c
\bigm\| \bigm\| \sigma rhg(w)\bigm\| \bigm\| Lp),\theta (\BbbT ,\omega ) \leq
\leq c \mathrm{s}\mathrm{u}\mathrm{p}
h\leq \delta
\bigm\| \bigm\| \sigma rhg(w)\bigm\| \bigm\| Lp),\theta (\BbbT ,\omega ) = \Omega r(g, \cdot )p),\theta ,\omega . (10)
Hence, by (5) and (10), we obtain
\Omega r
\bigl(
g+, \cdot
\bigr)
p),\theta ,\omega
\leq c
\Bigl\{
\Omega r
\bigl(
g, \cdot
\bigr)
p),\theta ,\omega
+\Omega r
\bigl(
S\BbbT (g), \cdot
\bigr)
p),\theta ,\omega
\Bigr\}
\leq c\Omega r
\bigl(
g, \cdot )p),\theta ,\omega .
Lemma 2 is proved.
Lemma 3 [13]. Let g \in \scrE p),\theta (\BbbU , \omega ), \omega \in Ap(\BbbT ), 1 < p < \infty , and \theta > 0. If
\sum n
k=0
\gamma k(g)w
k
is the nth partial sum of Taylor series of g at the origin, then there exists a positive constant c
independent of n = 1, 2, . . . such that\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| g(w) -
n\sum
k=0
\gamma k(g)w
k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp),\theta (\BbbT ,\omega )
\leq c\Omega r
\biggl(
g,
1
n
\biggr)
p),\theta ,\omega
, r = 1, 2, 3, . . . .
3. Proof of Theorem 1. Let \omega \in Ap(\Gamma ), \omega 0 \in Ap(\BbbT ), \omega 1 \in Ap(\BbbT ), 1 < p < \infty , \theta > 0. Let
\Gamma := \Gamma 1 \cup \Gamma -
2 , where \Gamma 1,\Gamma 2 \in \scrS and f \in \scrE p),\theta (G,\omega ). We get\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma ,\omega )
\leq
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta
\bigl(
\Gamma 1,\omega
\bigr) + \bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma 2,\omega )
.
Since f \in \scrE p),\theta (G,\omega ) we have f0 \in \scrL p),\theta
\bigl(
\Gamma 1, \omega
\bigr)
and f1 \in \scrL p),\theta (\Gamma 2, \omega ). For \zeta \in \Gamma 1 and
\xi \in \Gamma 2, by means of (3), (4) and (6), we have
f(\zeta ) =
\bigl[
f+0
\bigl(
\varphi (\zeta )
\bigr)
- f - 0
\bigl(
\varphi (\zeta )
\bigr)
]
\bigl(
\varphi \prime (\zeta )
\bigr) 1/(p - \varepsilon )
(11)
and
f(\xi ) =
\Bigl[
f+1
\bigl(
\varphi 1(\xi )
\bigr)
- f - 1
\bigl(
\varphi 1(\xi )
\bigr) \Bigr] \bigl(
\varphi 1(\xi )
\bigr) - 2/(p - \varepsilon )\bigl(
\varphi \prime
1(\xi )
\bigr) 1/(p - \varepsilon )
. (12)
It suffices to prove validity the inequalities
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta
\bigl(
\Gamma 1,\omega
\bigr) \leq c
\Biggl\{
\Omega r
\biggl(
f0,
1
n
\biggr)
p),\theta ,\omega 0
+\Omega r
\biggl(
f1,
1
n
\biggr)
p),\theta ,\omega 1
\Biggr\}
(13)
and
\| f - Rn(f)\| Lp),\theta (\Gamma 2,\omega )
\leq c
\Biggl\{
\Omega r
\biggl(
f0,
1
n
\biggr)
p),\theta ,\omega 0
+\Omega r
\biggl(
f1,
1
n
\biggr)
p),\theta ,\omega 1
\Biggr\}
. (14)
Firstly, we prove the estimation (13). Let us take a z\prime \in G -
1 . Then by relations (7) and (11) we
have
n\sum
k=0
akFk,p,\varepsilon (z
\prime ) =
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
n\sum
k=0
ak
\bigl[
\varphi (z\prime )
\bigr] k
+
+
1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
\sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k
\zeta - z\prime
d\zeta =
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APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 973
=
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
n\sum
k=0
ak
\bigl[
\varphi
\bigl(
z\prime
\bigr) \bigr] k
+
+
1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
\sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k
\zeta - z\prime
d\zeta -
- 1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
f+0
\bigl(
\varphi (\zeta )
\bigr)
\zeta - z\prime
d\zeta +
+
1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
f - 0
\bigl(
\varphi (\zeta )
\bigr)
\zeta - z\prime
d\zeta +
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z\prime
d\zeta .
Since [\varphi \prime (\zeta )]1/(p - \varepsilon ) f - 0
\bigl(
\varphi (\zeta )
\bigr)
\in E1(G -
1 ), we get
-
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
f - 0
\bigl(
\varphi (z\prime )
\bigr)
=
1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
f - 0
\bigl(
\varphi (\zeta )
\bigr)
\zeta - z\prime
d\zeta ,
therefore,
n\sum
k=0
akFk,p,\varepsilon (z
\prime ) =
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
n\sum
k=0
ak
\bigl[
\varphi (z\prime )
\bigr] k -
- 1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
\Bigl[
f+0
\bigl(
\varphi (\zeta )
\bigr)
-
\sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k\Bigr]
\zeta - z\prime
d\zeta -
-
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
f - 0
\bigl(
\varphi (z\prime )
\bigr)
+
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z\prime
d\zeta . (15)
If z\prime \in G -
2 , then by the relations (2) and (12) we have
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z
\prime ) = - 1
2\pi i
\int
\Gamma 2
\bigl[
\varphi 1(\xi )
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(\xi )
\bigr] - 2/(p - \varepsilon )
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k
\xi - z\prime
d\xi =
=
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi 1(\xi )
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(\xi )
\bigr] - 2/(p - \varepsilon )
\Bigl(
f+1
\bigl(
\varphi 1(\xi )
\bigr)
-
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k\Bigr)
\xi - z\prime
d\xi =
=
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi 1(\xi )
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(\xi )
\bigr] - 2/(p - \varepsilon )
f - 1
\bigl(
\varphi 1(\xi )
\bigr)
\xi - z\prime
d\xi - 1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z\prime
d\xi
and by using Cauchy integral formula for
\bigl(
\varphi 1(\xi )
\bigr) - 2/(p - \varepsilon )\bigl(
\varphi \prime
1(\xi )
\bigr) 1/(p - \varepsilon )
f - 1
\bigl(
\varphi 1(\xi )
\bigr)
\in E1(G2), we
obtain
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974 A. TESTICI
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z
\prime ) = - 1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z\prime
d\xi +
+
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi 1(\xi )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\Bigl(
f+1
\bigl(
\varphi 1(\xi )
\bigr)
-
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k\Bigr)
\xi - z\prime
d\xi . (16)
Then, for z\prime \in G -
1 , by (15), (16) and (9), we get
n\sum
k=0
akFk,p,\varepsilon (z
\prime ) +
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z
\prime ) =
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
n\sum
k=0
ak
\bigl[
\varphi (z\prime )
\bigr] k -
- 1
2\pi i
\int
\Gamma 1
\bigl[
\varphi \prime (\zeta )
\bigr] 1/(p - \varepsilon )
\Bigl[
f+0
\bigl(
\varphi (\zeta )
\bigr)
-
\sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k\Bigr]
\zeta - z\prime
- d\zeta -
-
\bigl[
\varphi \prime (z\prime )
\bigr] 1/(p - \varepsilon )
f - 0
\bigl(
\varphi (z\prime )
\bigr)
+
+
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi 1(\xi )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\Bigl(
f+1
\bigl(
\varphi 1(\xi )
\bigr)
-
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k\Bigr)
\xi - z\prime
d\xi .
Now taking the limit as z\prime \rightarrow z \in \Gamma 1 along all nontangential paths outside \Gamma 1, we obtain
f(z) -
n\sum
k=0
akFk,p,\varepsilon (z) -
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z) =
=
\bigl[
\varphi \prime (z)
\bigr] 1/(p - \varepsilon )
\Biggl[
f+0 (\varphi (z)) -
n\sum
k=0
ak
\bigl[
\varphi (z)
\bigr] k\Biggr] -
- 1
2
\bigl[
\varphi \prime (z)
\bigr] 1/(p - \varepsilon )
\Biggl[
f+0 (\varphi (z)) -
n\sum
k=0
ak
\bigl[
\varphi (z)
\bigr] k\Biggr]
+
+ S\Gamma 1
\Biggl[ \bigl[
\varphi \prime \bigr] 1/(p - \varepsilon )
\Biggl( \bigl(
f+0 \circ \varphi
\bigr)
-
n\sum
k=0
ak [\varphi ]
k
\Biggr) \Biggr]
(z) -
- 1
2\pi i
\int
\Gamma 2
\bigl[
\varphi 1(\xi )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\Bigl( \sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k - f+1
\bigl(
\varphi 1(\xi )
\bigr) \Bigr)
\xi - z
d\xi (17)
a.e. on \Gamma 1. Since \omega \in Ap(\Gamma ), applying Theorem A for \Gamma 1 and using (17) and the Minkowski
inequality, we have
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma 1,\omega )
\leq c
\left\{
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f+0 (w) -
n\sum
k=0
akw
k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp),\theta (\BbbT ,\omega 0)
+
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f+1 (w) -
n\sum
k=1
\widetilde akwk
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp),\theta (\BbbT ,\omega 1)
\right\} .
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APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 975
The Faber coefficients ak and \widetilde ak are Taylor coefficients of f+0 and f+1 , respectively, at origin. Since
\omega 0 \in Ap(\BbbT ) and \omega 1 \in Ap(\BbbT ), by using Lemmas 3 and 2, we get
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma 1,\omega )
\leq c
\Biggl\{
\Omega r
\biggl(
f0,
1
n
\biggr)
p),\theta ,\omega 0
+\Omega r
\biggl(
f1,
1
n
\biggr)
p),\theta ,\omega 1
\Biggr\}
. (18)
Let z\prime \prime \in G2. Then by (8) and (12) we obtain
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z
\prime \prime ) =
\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(z
\prime \prime )
\bigr] - 2
p - \varepsilon
n\sum
k=1
\widetilde ak \bigl[ \varphi 1(z
\prime \prime )
\bigr] k -
- 1
2\pi i
\int
\Gamma 2
\bigl[
\varphi \prime
1(\xi )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k
\xi - z\prime \prime
d\xi =
=
\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(z)
\bigr] - 2
p - \varepsilon
n\sum
k=1
\widetilde ak \bigl[ \varphi 1(z
\prime \prime )
\bigr] k
+
+
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi \prime
1(\xi )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\Bigl(
f+1
\bigl(
\varphi 1(\xi )
\bigr)
-
\sum n
k=1 \widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k\Bigr)
\xi - z\prime \prime
d\xi -
- 1
2\pi i
\int
\Gamma 2
\bigl[
\varphi \prime
1(\xi )
\bigr] 1
p - \varepsilon
\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon f - 1
\bigl(
\varphi 1(\xi )
\bigr)
\xi - z\prime \prime
d\xi - 1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z\prime \prime
d\xi .
Since
\bigl(
\varphi 1(\xi )
\bigr) - 2/(p - \varepsilon )
(\varphi \prime
1(\xi ))
1/(p - \varepsilon ) f - 1
\bigl(
\varphi 1(\xi )
\bigr)
\in E1(G2), we have\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1/(p - \varepsilon ) \bigl[
\varphi 1
\bigl(
z\prime \prime
\bigr) \bigr] - 2/(p - \varepsilon )
f - 1
\bigl(
\varphi 1(z
\prime \prime )
\bigr)
=
=
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi \prime
1(\xi )
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(\xi )
\bigr] - 2/(p - \varepsilon )
f - 1
\bigl(
\varphi 1(\xi )
\bigr)
\xi - z\prime \prime
d\xi .
This equality implies that
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z
\prime \prime ) =
\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1/(p - \varepsilon ) \bigl[
\varphi 1(z
\prime \prime )
\bigr] - 2
p - \varepsilon
n\sum
k=1
\widetilde ak \bigl[ \varphi 1(z
\prime \prime )
\bigr] k
+
+
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi \prime
1(\xi )
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\Bigl(
f+1
\bigl(
\varphi 1(\xi )
\bigr)
-
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k\Bigr)
\xi - z\prime \prime
d\xi -
-
\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1/(p - \varepsilon ) \bigl[
\varphi 1
\bigl(
z\prime \prime
\bigr) \bigr] - 2
p - \varepsilon f - 1
\bigl(
\varphi 1(z
\prime \prime )
\bigr)
- 1
2\pi i
\int
\Gamma 2
f(\xi )
\xi - z\prime \prime
d\xi . (19)
Let z\prime \prime \in G1. Then by (1) and (11) we get
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
976 A. TESTICI
n\sum
k=0
akFk,p,\varepsilon
\bigl(
z\prime \prime
\bigr)
=
1
2\pi i
\int
\Gamma 1
(\varphi \prime (\zeta ))
1
p - \varepsilon
\sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k
\zeta - z\prime \prime
d\zeta =
=
1
2\pi i
\int
\Gamma 1
(\varphi \prime (\zeta ))
1
p - \varepsilon
\Bigl( \sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k - f+0
\bigl(
\varphi (\zeta )
\bigr) \Bigr)
\zeta - z\prime \prime
d\zeta +
+
1
2\pi i
\int
\Gamma 1
\bigl(
\varphi \prime (\zeta )
\bigr) 1
p - \varepsilon f - 0 (\varphi (\zeta ))
\zeta - z\prime \prime
d\zeta +
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z\prime \prime
d\zeta ,
and by using the Cauchy integral formula for
\bigl(
\varphi \prime (\zeta )
\bigr) 1
p - \varepsilon f - 0 (\varphi (\zeta )) \in E1(G -
1 ), we obtain
n\sum
k=0
akFk,p,\varepsilon
\bigl(
z\prime \prime
\bigr)
=
1
2\pi i
\int
\Gamma 1
f(\zeta )
\zeta - z\prime \prime
d\zeta +
+
1
2\pi i
\int
\Gamma 1
\bigl(
\varphi \prime (\zeta )
\bigr) 1
p - \varepsilon
\Bigl( \sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k - f+0
\bigl(
\varphi (\zeta )
\bigr) \Bigr)
\zeta - z\prime \prime
d\zeta . (20)
For z\prime \prime \in G2, the relations (19), (20) and (9) imply that
n\sum
k=0
akFk,p,\varepsilon
\bigl(
z\prime \prime
\bigr)
+
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z
\prime \prime ) =
=
\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1/(p - \varepsilon ) \bigl[
\varphi 1
\bigl(
z\prime \prime
\bigr) \bigr] - 2
p - \varepsilon
n\sum
k=1
\widetilde ak \bigl[ \varphi 1(z
\prime \prime )
\bigr] k
+
+
1
2\pi i
\int
\Gamma 2
\bigl[
\varphi \prime
1(\xi )
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(\xi )
\bigr] - 2
p - \varepsilon
\Bigl(
f+1
\bigl(
\varphi 1(\xi )
\bigr)
-
\sum n
k=1
\widetilde ak\bigl[ \varphi 1(\xi )
\bigr] k\Bigr)
\xi - z\prime \prime
d\xi -
-
\bigl[
\varphi \prime
1(z
\prime \prime )
\bigr] 1/(p - \varepsilon ) \bigl[
\varphi 1
\bigl(
z\prime \prime
\bigr) \bigr] - 2
p - \varepsilon f - 1
\bigl(
\varphi 1(z
\prime \prime )
\bigr)
+
+
1
2\pi i
\int
\Gamma 1
\bigl(
\varphi \prime (\zeta )
\bigr) 1
p - \varepsilon
\Bigl( \sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k - f+0
\bigl(
\varphi (\zeta )
\bigr) \Bigr)
\zeta - z\prime \prime
d\zeta .
Taking the limit as z\prime \prime \rightarrow z \in \Gamma 2 along all nontangential paths inside \Gamma 2, by (11) we obtain
f(z) -
n\sum
k=0
akFk,p,\varepsilon (z) -
n\sum
k=1
\widetilde ak \widetilde Fk,p,\varepsilon (1/z) =
=
\bigl[
\varphi \prime
1(z)
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(z)
\bigr] - 2
p - \varepsilon f+1 (\varphi 1(z)) -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
APPROXIMATION BY RATIONAL FUNCTIONS ON DOUBLY CONNECTED DOMAINS . . . 977
- 1
2
\bigl[
\varphi \prime
1(z)
\bigr] 1/(p - \varepsilon )\bigl[
\varphi 1(z)
\bigr] - 2
p - \varepsilon
\Biggl[
n\sum
k=1
\widetilde ak\bigl[ \varphi 1(z)
\bigr] k - f+1
\bigl(
\varphi 1(z)
\bigr) \Biggr]
-
- S\Gamma 2
\Biggl[ \bigl[
\varphi \prime
1
\bigr] 1/(p - \varepsilon )
[\varphi 1]
- 2
p - \varepsilon
\Biggl(
n\sum
k=1
\widetilde ak [\varphi 1]
k -
\bigl(
f+1 \circ \varphi 1
\bigr) \Biggr) \Biggr]
(z) -
- 1
2\pi i
\int
\Gamma 1
\bigl(
\varphi \prime (\zeta )
\bigr) 1
p - \varepsilon
\Bigl( \sum n
k=0
ak
\bigl[
\varphi (\zeta )
\bigr] k - f+0
\bigl(
\varphi (\zeta )
\bigr) \Bigr)
\zeta - z
d\zeta (21)
a.e. on \Gamma 2. Since \omega \in Ap(\Gamma ), applying Theorem A for \Gamma 2 and by using (17) and Minkowski
inequality, we obtain
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma 2,\omega )
\leq c
\left\{
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f+0 (w) -
n\sum
k=0
akw
k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp),\theta (\BbbT ,\omega 0)
+
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f+1 (w) -
n\sum
k=1
\widetilde akwk
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp),\theta (\BbbT ,\omega 1)
\right\} .
The Faber coefficients ak and \widetilde ak are Taylor coefficients of f+0 and f+1 , respectively, at origin. Since
\omega 0 \in Ap(\BbbT ) and \omega 1 \in Ap(\BbbT ), finally by using Lemmas 3 and 2, we have
\bigm\| \bigm\| f - Rn(f)
\bigm\| \bigm\|
Lp),\theta (\Gamma 2,\omega )
\leq c
\Biggl\{
\Omega r
\biggl(
f0,
1
n
\biggr)
p),\theta ,\omega 0
+\Omega r
\biggl(
f1,
1
n
\biggr)
p),\theta ,\omega 1
\Biggr\}
. (22)
Hence, (18) and (22) complete the proof of Theorem 1.
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Received 09.12.18,
after revision — 09.03.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
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| id | umjimathkievua-article-559 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:03Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/96/f971a98ccc07d6be6aa3d1068ee11c96.pdf |
| spelling | umjimathkievua-article-5592025-03-31T08:47:53Z Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes Approximation By Rational Functions On Doubly Connected Domains In Weighted Generalized Grand Smirnov Classes Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes Testici, A. Testici, Ahmet Testici, A. Doubly connected domai modulus of smoothnes Faber-Laurent serie generalized grand Smirnov clas Carleson curve Doubly connected domai modulus of smoothnes Faber-Laurent serie generalized grand Smirnov clas Carleson curve UDC 517.5 Let $G\subset \mathbb{C}$ be a doubly connected domain bounded by two rectifiable Carleson curves.&nbsp;In this work, we use the higher modulus of smoothness in order to investigate the approximation properties of $(p-\varepsilon)$-Faber–Laurent rational functions in the subclass of weighted generalized grand Smirnov classes ${E}^{p),\theta } ( {G,\omega })$ of analytic functions. УДК 517.5 Наближення рацiональними функцiями для двозв’язних областей у зважених узагальнених великих класах Смiрнова Нехай $G ⊂ ℂ$— двозв’язна область обмежена двома спрямними кривими Карлесона. У цiй роботi за допомогою вищого модуля гладкостi вивчається апроксимацiйнi властивостi рацiональних $p - ε$ функцiй Фабера –Лорана у пiдкласах зважених узагальнених великих класiв Смiрнова $E^{p),θ}(G,ω)$ аналiтичних функцiй. &nbsp; Institute of Mathematics, NAS of Ukraine 2021-07-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/559 10.37863/umzh.v73i7.559 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 7 (2021); 964 - 978 Український математичний журнал; Том 73 № 7 (2021); 964 - 978 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/559/9072 Copyright (c) 2021 Ahmet Testici |
| spellingShingle | Testici, A. Testici, Ahmet Testici, A. Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title | Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title_alt | Approximation By Rational Functions On Doubly Connected Domains In Weighted Generalized Grand Smirnov Classes Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title_full | Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title_fullStr | Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title_full_unstemmed | Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title_short | Approximation by rational functions on doubly connected domains in weighted generalized grand Smirnov classes |
| title_sort | approximation by rational functions on doubly connected domains in weighted generalized grand smirnov classes |
| topic_facet | Doubly connected domai modulus of smoothnes Faber-Laurent serie generalized grand Smirnov clas Carleson curve Doubly connected domai modulus of smoothnes Faber-Laurent serie generalized grand Smirnov clas Carleson curve |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/559 |
| work_keys_str_mv | AT testicia approximationbyrationalfunctionsondoublyconnecteddomainsinweightedgeneralizedgrandsmirnovclasses AT testiciahmet approximationbyrationalfunctionsondoublyconnecteddomainsinweightedgeneralizedgrandsmirnovclasses AT testicia approximationbyrationalfunctionsondoublyconnecteddomainsinweightedgeneralizedgrandsmirnovclasses |