Polynomial approximation in Bergman spaces
The purpose of this work is to obtain Jackson and converse inequalities of the polynomial approximation in Bergman spaces. Some known results presented for the moduli of continuity are extended to the moduli of smoothness. We proved some simultaneous approximation theorems and obtained the Nikolskii...
Збережено в:
| Дата: | 2016 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1850 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507725398540288 |
|---|---|
| author | Akgün, R. Акгюн, Р. |
| author_facet | Akgün, R. Акгюн, Р. |
| author_sort | Akgün, R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:29:54Z |
| description | The purpose of this work is to obtain Jackson and converse inequalities of the polynomial approximation in Bergman spaces.
Some known results presented for the moduli of continuity are extended to the moduli of smoothness. We proved some
simultaneous approximation theorems and obtained the Nikolskii – Stechkin inequality for polynomials in these spaces. |
| first_indexed | 2026-03-24T02:13:53Z |
| format | Article |
| fulltext |
UDC 517.5
R. Akgün (Balikesir Univ., Turkey)
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES*
ПОЛIНОМIАЛЬНI НАБЛИЖЕННЯ У ПРОСТОРАХ БЕРГМАНА
The purpose of this work is to obtain Jackson and converse inequalities of the polynomial approximation in Bergman spaces.
Some known results presented for the moduli of continuity are extended to the moduli of smoothness. We proved some
simultaneous approximation theorems and obtained the Nikolskii – Stechkin inequality for polynomials in these spaces.
Meтою даної роботи є встановлення нерiвностi Джексона та обернених нерiвностей для полiномiальних наближень
у просторi Бергмана. Деякi вiдомi результати для модулiв неперервностi узагальнено на модулi гладкостi. Доведено
деякi спiльнi теореми про наближення та встановлено нерiвнiсть Нiкольського – Стечкiна для полiномiв у цих
просторах.
1. Statement of problem. Let \Omega \subset \BbbC be an arbitrary domain in the complex plane. The Bergman
space Bp (\Omega ) consists of all functions f analytic in \Omega for which
\| f\| p :=
\left\{
\int \int
\Omega
| f (z)| p d\sigma (z)
\right\}
1/p
< \infty
for 0 < p < \infty , where d\sigma (z) :=
1
\pi
dxdy is area measure on \Omega . B\infty (\Omega ) is the set of functions
f bounded and analytic in \Omega . In this case we set \| f\| \infty := \mathrm{s}\mathrm{u}\mathrm{p}z\in \Omega | f (z)| . B0 (\Omega ) is the set of
functions f analytic in \Omega with
\| f\| 0 := \mathrm{e}\mathrm{x}\mathrm{p}
\left( \int \int
\Omega
\mathrm{l}\mathrm{o}\mathrm{g} | f (z)| d\sigma (z)
\right) < \infty ,
where \mathrm{l}\mathrm{o}\mathrm{g} | f | is summable on \Omega . \| f\| p is called the norm of f and it is a true norm if p \geq 1. If \Omega is
the complex unit disc \BbbD , then we will write Bp instead of Bp (\BbbD ) .
For a function f, analytic in \BbbD , the integral means are defined by
Mp (r, f) :=
\left\{ 1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| f \Bigl( rei\theta \Bigr) \bigm| \bigm| \bigm| p d\theta
\right\}
1/p
, 0 < p < \infty , 0 \leq r < 1,
and M\infty (r, f) := \mathrm{s}\mathrm{u}\mathrm{p}\theta \in [0,2\pi )
\bigm| \bigm| f \bigl( rei\theta \bigr) \bigm| \bigm| . If they are stay bounded as r \rightarrow 1 - , then f is said to
belong to Hardy space Hp. Thus H\infty = B\infty consists of all bounded analytic functions in \BbbD . The
norm \| f\| Hp of a function f \in Hp is defined as the limit of Mp (r, f) as r \rightarrow 1 - . It is a true norm
if p \geq 1. H0 is the set of functions f analytic in \BbbD with
* This paper was supported by Balikesir University (Scientific Research Project 1.2015.0015).
c\bigcirc R. AKGÜN, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4 435
436 R. AKGÜN
\| f\| H0 := \mathrm{e}\mathrm{x}\mathrm{p}
\left( 1
2\pi
2\pi \int
0
\mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| \bigm| f \Bigl( ei\theta \Bigr) \bigm| \bigm| \bigm| d\theta
\right) < \infty ,
where \mathrm{l}\mathrm{o}\mathrm{g} | f | is summable on \BbbD . It is well known that Hp \subset B2p and \| f\| 2p \leq \| f\| Hp
for 0 < p < \infty .
The main initial problem in Approximation Theory is density of the set of polynomials in spaces
investigated. The first results related with the density of the set of polynomials in Bergman spaces
were obtained in 1934 by Farrel and Markushevich (see, e.g., [6, 17]). Also the same problems was
considered by Al’per [1], Burbea [3] and Metzger [9] for Bergman spaces Bp (\Omega ) , where p \geq 1 and
\Omega is a bounded simply connected region with a simply connected complement. In [1] Al’per defined
a moduli of continuity in Bp (\Omega ) and proved the direct theorem in this space. Also in the same
article Al’per also obtained the Quade type [10] converse inequalities in Bp (\Omega ). Later L. F. Zhong
[25] construct a polynomial that near best approximant for functions of Bergman spaces Bp (\Omega ) ,
where 1 < p < \infty and \Omega is a bounded region with a sufficiently smooth boundary. For discs in \BbbC
these problems in Bp was considered by many mathematicians: For p = \infty Storozhenko [14] proved
a Jackson-type direct theorem in B\infty for boundary moduli of smoothness. She also defined [15]
different moduli of smoothness on the unit circle and obtained a direct theorem of approximation in
B\infty . Later Kryakin [7] extended the properties of Storozhenko’s moduli of smoothness on whole
\BbbD and he find Jackson-type direct theorem, simultaneous approximation theorem in B\infty . For 0 <
< p \leq \infty F. Ch. Xing and C. L. Su [23] (p \geq 1) and X. C. Shen and F. Ch. Xing [13] (0 < p < 1)
was proved direct theorem for moduli of continuity in Bp. Also for 0 < p \leq \infty X. C. Shen and
F. Ch. Xing [13] (0 < p < 1) and F. Ch. Xing and Z. Su [24] (p \geq 1) proved Quade-type converse
inequalities in Bp. A different method has been applied by G. Ren and M. Wang [11] to obtain
Jackson inequality with moduli of continuity in Bp (\Omega R) , 0 < p \leq \infty , where \Omega R is the arbitrary
disc with radius R. For 1 \leq p \leq \infty , M. Sh. Shabozov and O. Sh. Shabozov [12] find some exact
constants of Jackson inequality with first and second degree moduli of smoothness in Bp. Also
several questions related with the approximation by algebraic polynomials were considered by S. B.
Vakarchuk [18] in Bp, 1 \leq p < \infty . For p > 0, F. Ch. Xing [21] proved the Bernstein inequality and
Quade-type converse inequalities [22] in Bp. For 0 < p \leq \infty , M. Z. Wang and G. Ren [19] proved
the direct theorem on polynomial approximation in Bp (\Omega R).
Above results are contain inequalities with moduli of continuity, except the results of Storozhenko
[14] and Kryakin [7]. In this work we will generalize these results to the moduli of smoothness of
arbitrary order. For 0 < p \leq \infty , we will prove some direct and converse theorems of polynomial
approximation in Bp for moduli of smoothness. In case of 1 < p < \infty , we obtain simultaneous
approximation theorem in Bp. The rests of the work organized as follows. Section 2 contains the
main properties of the moduli of smoothness of functions in Bp, 0 < p \leq \infty . In Section 3 we give a
proof of first and second-type Jackson’s direct theorems for Bp, 0 < p \leq \infty . In Section 4 we prove
simultaneous theorems of polynomial approximation in Bp, 1 < p < \infty . In Section 5 using Szego
composition we prove a Nikolskii – Stechkin inequality in Bp, 0 \leq p \leq \infty . Then we obtain converse
theorems of polynomial approximation in Bp, 0 < p \leq \infty . Throughout the work we will denote
by c and C positive constants which are different in different occurrences. Let \BbbN = \{ 1, 2, 3, . . .\} ,
\BbbN 0 = \{ 0\} \cup \BbbN , \BbbZ = \{ 0,\pm 1,\pm 2,\pm 3, . . .\} . By j = 0,m we will mean j = 0, 1, 2, 3, . . . ,m.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES 437
2. Moduli of smoothness. Let z, v \in \BbbC and let f be a function defined on the points zj \in \BbbC ,
j \in \BbbN 0. The divided difference of function f at the points zj , j \in \BbbN 0, is defined as
[z0, z1, . . . , zm]f :=
m\sum
j=0
f (zj)
\prod
i \not =j
(zj - zi)
- 1 . (1)
Choosing zj = zvj , j \in \BbbN 0, we define finite difference operator for m \in \BbbN
[z, f ]0v := f (z) , [z, f ]mv := [z, zv, . . . , zvm]f
m\prod
j=1
\bigl(
z - zvj
\bigr)
.
From (1) we can see that
[z, f ]mv =
m\sum
j=0
( - 1)j Pj,m
\bigl(
v - 1
\bigr)
f
\bigl(
zvj
\bigr)
,
where
Pj,m (z) :=
zj(j - 1)/2
\bigl(
zj+1 - 1
\bigr)
. . . (zm - 1)
(z - 1) . . . (zm - j - 1)
, j = 0,m - 1,
Pm,m (z) := zm(m - 1)/2.
We give the basic properties of finite difference operator: Let z, v \in \BbbC and m,n \in \BbbN . Then
[z, f ]mv = [z, f ]m - 1
v - v - (m - 1) [zv, f ]m - 1
v , (2)
[z, f ]mv =
1\sum
j1=0
. . .
1\sum
jm=0
( - 1)
\sum m
l=1 jl v -
\sum m
l=1(l - 1)jlf
\Bigl(
zv
\sum m
l=1 jl
\Bigr)
, (3)
[z, f ]mvn =
n - 1\sum
j1=0
. . .
n - 1\sum
jm=0
v -
\sum m
l=1(l - 1)jl
\Bigl[
zv
\sum m
l=1 jl , f
\Bigr] m
v
. (4)
If f is analytic in \BbbD , and | z| < 1, | v| \leq 1, i = 0,m - 1, we have
[z, f ]mv = zm - i
1\int
v
um - i - 1
1 du1
1\int
v
um - i - 2
2 du2 . . .
1\int
v
\Bigl[
zu1 . . . um - i, f
(m - i)
\Bigr] i
v
dum - i. (5)
Here f (i) (z) =
dif (z)
dzi
is the ith derivative of f(z).
We define generalized finite difference operator [z, f ]m,s
v by
[z, f ]0,sv := f (z) , [z, f ]m,s
v := [z, f ]m - 1,s
v - v - (m+s - 1) [zv, f ]m - 1,s
v .
By (2) we have [z, f ]m,0
v = [z, f ]mv . On the other hand
[z, f ]m,s
v =
1\sum
j1=0
. . .
1\sum
jm=0
( - 1)
\sum m
l=1 jl v -
\sum m
l=1(s+l - 1)jlf
\Bigl(
zv
\sum m
l=1 jl
\Bigr)
.
If k,m \in \BbbN 0 and s \in \BbbZ , then
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
438 R. AKGÜN
d
dz
([z, f ]m,s
v ) =
\biggl[
z,
d
dz
f
\biggr] m,s - 1
v
and [z, f ]m+k
v =
\Bigl[
z, [\cdot , f ]m,k
v
\Bigr] k
v
. (6)
All these properties can be proved by induction (see [7, 8, 15]). We define the moduli of smoothness
as the following: Let f \in Bp, 0 < p \leq \infty and \delta \geq 0. The moduli of smoothness of order m \in \BbbN is
defined as
\~\omega m (\delta , f)p := \mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq \delta
\| [\cdot , f ]meit\| p , 0 < t \leq 2\pi . (7)
This moduli was first defined, on the unit circle \BbbT , by Storozhenko [15] with z = ei\varphi and then
Kryakin [7] proved its above properties on \BbbD . Letting m = 1 and z = ei\varphi in (7) this moduli
coincides, on \BbbT , with ordinary boundary moduli of continuity. If m > 1, then \~\omega m (\cdot , f)p and
ordinary boundary moduli of smoothness are different.
It is easy to see that if f, g \in Bp, 0 < p \leq \infty and 0 \leq \delta \leq \eta , then, there exists a constant c > 0,
depending only on m and p, such that
\~\omega m (\cdot , f)p \leq c \| f\| p and 0 = \~\omega m (0, f)p \leq \~\omega m (\delta , f)p \leq \~\omega m (\eta , f)p ,
\~\omega m (\cdot , f + g)p \leq \~\omega m (\cdot , f)p + \~\omega m (\cdot , f)p .
Let 0 < p \leq \infty and r \in \BbbN . We denote by Bp
r the class of functions f \in Bp having the property
f (r) \in Bp.
Theorem 1. Let f \in Bp, 0 < p \leq \infty , m, n \in \BbbN , \delta \geq 0 and s := \mathrm{m}\mathrm{i}\mathrm{n} \{ 1, p\} . Then there exists
a constant c > 0, depending only on p, such that
\~\omega m (n\delta , f)p \leq cnm - 1+(1/s)\~\omega m (\delta , f)p .
Proof. Using (4) we have
\| [z, f ]meint\| p \leq
\left\{
\int \int
\BbbD
\left[ n - 1\sum
j1=0
. . .
n - 1\sum
jm=0
\bigm| \bigm| \bigm| \Bigl[ zeit(\sum m
l=1 jl), f
\Bigr] m
eit
\bigm| \bigm| \bigm|
\right] p
d\sigma (z)
\right\}
1/p
:= I1/p.
For p \geq 1, s = 1 we obtain
I1/p \leq 2(p - 1)/p
n - 1\sum
j1=0
. . .
n - 1\sum
jm=0
\bigm\| \bigm\| \bigm\| \Bigl[ zeit(\sum m
l=1 jl), f
\Bigr] m
eit
\bigm\| \bigm\| \bigm\|
p
=
= 2(p - 1)/p
n - 1\sum
j1=0
. . .
n - 1\sum
jm=0
\| [z, f ]meit\| p \leq 2(p - 1)/pnm \| [z, f ]meit\| p =
= 2(p - 1)/pnm - 1+(1/s) \| [z, f ]meit\| p
and hence \~\omega m (n\delta , f)p \leq cnm - 1+(1/s)\~\omega m (\delta , f)p for p \geq 1. Now we suppose 0 < p < 1 and s = p.
Then
\| [z, f ]meint\| p \leq 2(1/p) - 1
n - 1\sum
j1=0
. . .
n - 1\sum
jm=0
\| [z, f ]meit\| p .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES 439
Therefore
\~\omega m (n\delta , f)p \leq nm+(1/s) - 1\~\omega m (\delta , f)p .
Theorem 1 is proved.
Theorem 2. Let 0 < p < 1, k \in \BbbN and f \in Bp
k . Then there exist a constant c > 0 and a
number t0 > 0, depending only on p, k, such that\bigm\| \bigm\| \bigm\| [\cdot , f ]keit\bigm\| \bigm\| \bigm\|
p
\leq ctk
\bigm\| \bigm\| \bigm\| f (k)
\bigm\| \bigm\| \bigm\|
p
holds for all 0 < t \leq t0, where the constant c depends only on p.
Proof. We set 0 \leq r < R < 1 and define t0 := (R - r) /Ak, where A \geq 1 is some constant
depending only on p. Then for all 0 < t \leq t0 we have [7] (Lemma 1)
Mp
p
\Bigl(
r, [\cdot , f ]keit
\Bigr)
\leq cptkpMp
p
\Bigl(
R, f (k)
\Bigr)
.
Then
1\int
0
Mp
p
\Bigl(
r, [\cdot , f ]keit
\Bigr)
rdr \leq cptkpMp
p
\Bigl(
R, f (k)
\Bigr)
R,
\bigm\| \bigm\| \bigm\| [\cdot , f ]keit\bigm\| \bigm\| \bigm\| p
p
\leq cptkp
1\int
0
Mp
p
\Bigl(
R, f (k)
\Bigr)
RdR = cptkp
\bigm\| \bigm\| \bigm\| f (k)
\bigm\| \bigm\| \bigm\| p
p
.
Theorem 2 is proved.
Theorem 3. Let k \in \BbbN , 0 < p \leq \infty , f \in Bp
k, m \in \BbbN 0 and \delta \geq 0. Then there exist a constant
c > 0 and a number \delta 0 > 0, depending only on p, k, such that
\~\omega m+k (\delta , f)p \leq c\delta k\~\omega m
\Bigl(
\delta , f (k)
\Bigr)
p
holds for all 0 < \delta \leq \delta 0, where the constant c depends only on p.
Proof. Let p \geq 1. Using (5) and the generalized Minkowski inequality for | z| < 1 we get
\bigm\| \bigm\| \bigm\| [z, f ]k+m
eit
\bigm\| \bigm\| \bigm\|
p
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
eit
. . .
1\int
eit
\Bigl[
zu1 . . . uk, f
(k)
\Bigr] m
eit
uk - 1
1 . . . zkduk . . . du1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq
1\int
eit
. . .
1\int
eit
\bigm\| \bigm\| \bigm\| \Bigl[ zu1 . . . uk, f (k)
\Bigr] m
eit
\bigm\| \bigm\| \bigm\|
p
duk . . . du1 =
=
1\int
eit
. . .
1\int
eit
\bigm\| \bigm\| \bigm\| \Bigl[ z, f (k)
\Bigr] m
eit
\bigm\| \bigm\| \bigm\|
p
duk . . . du1 \leq tk
\bigm\| \bigm\| \bigm\| \Bigl[ z, f (k)
\Bigr] m
eit
\bigm\| \bigm\| \bigm\|
and hence \~\omega m+k (\delta , f)p \leq \delta k\~\omega m
\bigl(
\delta , f (k)
\bigr)
p
for p \geq 1. Now let 0 < p < 1, \delta 0 := 1/ (4Ak)
and 0 < t \leq \delta 0. Then using the second equality of (6), Theorem 2 and the first equality of (6),
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
440 R. AKGÜN
respectively, we have\bigm\| \bigm\| \bigm\| [\cdot , f ]k+m
eit
\bigm\| \bigm\| \bigm\|
p
=
\bigm\| \bigm\| \bigm\| \bigm\| \Bigl[ \cdot , [\cdot , f ]m,k
eit
\Bigr] k
eit
\bigm\| \bigm\| \bigm\| \bigm\|
p
\leq ctk
\bigm\| \bigm\| \bigm\| \bigm\| dkf (z)
dzk
\Bigl(
[\cdot , f ]m,k
eit
\Bigr) \bigm\| \bigm\| \bigm\| \bigm\|
p
=
= ctk
\bigm\| \bigm\| \bigm\| \bigm\| \biggl( \biggl[ \cdot , dkf (z)
dzk
\biggr] m
eit
\biggr) \bigm\| \bigm\| \bigm\| \bigm\|
p
= ctk
\bigm\| \bigm\| \bigm\| \Bigl( \Bigl[ \cdot , f (k)
\Bigr] m
eit
\Bigr) \bigm\| \bigm\| \bigm\|
p
and \~\omega m+k (\delta , f)p \leq c\delta k\~\omega m
\bigl(
\delta , f (k)
\bigr)
p
for 0 < p < 1.
Theorem 3 is proved.
Corollary 1. Let 0 < p \leq \infty , k \in \BbbN , f \in Bp
k and \delta \geq 0. Then there exist a constant c > 0 and
a number \delta 0 > 0, depending only on p, k, such that
\~\omega k (\delta , f)p \leq c\delta k
\bigm\| \bigm\| \bigm\| f (k)
\bigm\| \bigm\| \bigm\|
p
holds for all 0 < \delta \leq \delta 0, where the constant c depends only on p.
3. Direct theorems. In this section we will prove Jackson’s fist- and second-type direct theorems
in Bp, 0 < p \leq \infty . Let En (f)p := \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
\| f - Pn\| p : Pn \in \scrP n
\Bigr\}
for n \in \BbbN , 0 < p \leq \infty and f \in Bp.
Here \scrP n is the set of the algebraic polynomials of degree at most n.
Lemma 1 [14]. Let 0 < p < 1, F be an analytic function in \Omega R. Then there exists a constant
c > 0, depending only on p, such that\left( \pi \int
- \pi
\bigm| \bigm| F \bigl( rei\varphi \bigr) \bigm| \bigm| d\varphi
\right) p
\leq C (\rho - r)p - 1
\pi \int
- \pi
\bigm| \bigm| F \bigl( \rho ei\varphi \bigr) \bigm| \bigm| p d\varphi
hold for 0 \leq r < \rho < R. Furthermore, if f \in Hp, then the values \rho = R = 1 are assumed.
Jackson’s fist-type direct theorem is the following theorem.
Theorem 4. Let 0 < p \leq \infty , f \in Bp, m, n \in \BbbN and n > m. Then there exists a constant c > 0
independent of n such that
En (f)p \leq c\~\omega m (1/n, f)p . (8)
Proof. Let 0 < r < 1, \alpha > 0 and
K\alpha
n
\bigl(
reit
\bigr)
:=
1\biggl(
\alpha + n
n
\biggr) \bigl(
reit
\bigr) - n
(1 - reit)1+\alpha
be the kernels of (C - \alpha )-means of f \in Bp, 0 < p \leq \infty . In this case
1
2\pi
\pi \int
- \pi
K\alpha
n
\bigl(
reit
\bigr)
dt = 1.
We set
\sansP n (z) =:
m\sum
j=1
( - 1)j+1 1
2\pi
\pi \int
- \pi
Pj,m
\Bigl( \bigl(
reit
\bigr) - 1
\Bigr)
f
\bigl(
zrjeijt
\bigr)
K\alpha
n - m+1
\bigl(
reit
\bigr)
dt.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES 441
The expression \mathrm{P}n (z) is a polynomial [8] of degree at most n. We will prove that
\| f - \mathrm{P}n\| p \leq c\~\omega m (1/n, f)p . (9)
This implies (8). Let 0 < p < 1, \alpha >
m+ 2
p
- 1 and n1 := n - m+ 1. Then
| f (z) - \mathrm{P}n (z)| \leq
1
2\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\pi \int
- \pi
([z, f ]mreit)K
\alpha
n1
\bigl(
reit
\bigr)
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
1\biggl(
\alpha + n1
n1
\biggr) 1
2\pi
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\pi \int
- \pi
([z, f ]mreit)
\bigl(
reit
\bigr) - n1
\Biggl(
1 -
\bigl(
reit
\bigr) n1+1
1 - reit
\Biggr) 1+\alpha
dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq r - n1\biggl(
\alpha + n1
n1
\biggr) 1
2\pi
\pi \int
- \pi
| [z, f ]mreit |
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl(
1 -
\bigl(
reit
\bigr) n1+1
1 - reit
\Biggr) \bigm| \bigm| \bigm| \bigm| \bigm|
1+\alpha
dt.
Since the function F (\varsigma ) := [z, f ]m\varsigma
\biggl(
1 - \varsigma n+1
1 - \varsigma
\biggr) 1+\alpha
is analytic and belong to Hp, from Lemma 1
we obtain
| f (z) - \mathrm{P}n (z)| p \leq
Cr - n1p\biggl(
\alpha + n1
n1
\biggr) p
(1 - r)1 - p
\pi \int
- \pi
| [z, f ]meit |
p
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl(
1 -
\bigl(
eit
\bigr) n1+1
1 - eit
\Biggr) \bigm| \bigm| \bigm| \bigm| \bigm|
(1+\alpha )p
dt.
Integrating the last inequality with respect to z we get
\| f - \mathrm{P}n\| pp \leq
Cr - n1p\biggl(
\alpha + n1
n1
\biggr) p
(1 - r)1 - p
\pi \int
0
\~\omega p
m (t, f)p
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n}n1t/2
\mathrm{s}\mathrm{i}\mathrm{n} t/2
\bigm| \bigm| \bigm| \bigm| (1+\alpha )p
dt.
Taking r = 1 - (1/n1) we have (1 - r)p - 1 = (n1)
1 - p and
\biggl(
\alpha + n1
n1
\biggr) - p
\leq c (n1)
- \alpha p. Hence
\| f - \mathrm{P}n\| pp \leq C (n1)
1 - (\alpha +1)p
\pi \int
0
\~\omega p
m (t, f)p
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n}n1t/2
\mathrm{s}\mathrm{i}\mathrm{n} t/2
\bigm| \bigm| \bigm| \bigm| (1+\alpha )p
dt =
= C (n1)
1 - (\alpha +1)p
\left\{
1/n1\int
0
\~\omega p
m (t, f)p
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n}n1t/2
\mathrm{s}\mathrm{i}\mathrm{n} t/2
\bigm| \bigm| \bigm| \bigm| (1+\alpha )p
dt +
+
\pi \int
1/n1
\~\omega p
m (t, f)p
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n}n1t/2
\mathrm{s}\mathrm{i}\mathrm{n} t/2
\bigm| \bigm| \bigm| \bigm| (1+\alpha )p
dt
\right\} .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
442 R. AKGÜN
Therefore using the property \~\omega p
l (\lambda \delta , f)p \leq (\lambda + 1)l \~\omega p
l (\delta , f)p , \lambda , \delta > 0, of moduli we obtain
\| f - \mathrm{P}n\| pp \leq C (n1)
1 - (\alpha +1)p \~\omega p
m (1/n1, f)p
\left\{ (n1)
(\alpha +1)p
1/n1\int
0
dt +
+
\pi \int
1/n1
(n1t+ 1)m
\biggl(
t
\pi
\biggr) - (1+\alpha )p
dt
\right\} \leq c\~\omega m (1/n1, f)p \leq c\~\omega m (1/n, f)p .
Let p \geq 1 and \alpha > m. Using generalized Minkowski’s inequality we get
\| f - \mathrm{P}n\| p \leq
1\biggl(
\alpha + n1
n1
\biggr) \pi \int
- \pi
\| [\cdot , f ]meit\| p
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n}n1t/2
\mathrm{s}\mathrm{i}\mathrm{n} t/2
\bigm| \bigm| \bigm| \bigm| 1+\alpha
dt \leq
\leq C (n1)
- \alpha \~\omega m (1/n1, f)p
\left\{ (n1)
1+\alpha
1/n1\int
0
dt+ (n1)
m
\pi \int
1/n1
\biggl(
t
\pi
\biggr) - (1+\alpha )
dt
\right\} \leq
\leq c\~\omega m (1/n1, f)p \leq c\~\omega m (1/n, f)p
and (9) follows.
Theorem 4 is proved.
We note that the last theorem is a generalization of the results of Storozhenko [14, p. 207]
(Theorem 1), Kryakin [7, p. 26] (Theorem 1), Xing and Su [23] (1 \leq p < \infty ) and X. C. Shen and
F. Ch. Xing [13] (0 < p < 1), Kryakin and Trebels [8] (Theorem 2.3) (0 < p < \infty ), Ren and Wang
[11] (Theorem 3.5) (0 < p \leq \infty ), Ren and Wang [19] (Theorem 3.4) (0 < p \leq \infty ).
From Theorem 4 we have the following Jackson’s second-type direct theorem.
Theorem 5. Let m,n, k \in \BbbN , n > m+k, 0 < p \leq \infty and f \in Bp
k . Then there exists a constant
c > 0 independent of n such that
En (f)p \leq cn - k\~\omega m
\Bigl(
1/n, f (k)
\Bigr)
p
holds.
This theorem generalizes Theorem 3 of [7].
Corollary 2. Let k, n \in \BbbN , 0 < p \leq \infty and f \in Bp
k . Then there exists a constant c > 0
independent of n such that
En (f)p \leq cn - k
\bigm\| \bigm\| \bigm\| f (k)
\bigm\| \bigm\| \bigm\|
p
holds for n > k.
4. Simultaneous approximation. We suppose that f \in Bp, 0 < p \leq \infty , has Taylor expansion
\infty \sum
k=0
ck (f) z
k. (10)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES 443
Let Tn (f) := Tn (f, z) :=
\sum n
k=0 ck (f) z
k be the nth partial sum of (10). The operator Tn :
Bp \rightarrow Bp, f \rightarrow Tn (f, \cdot ) will be called the nth partial sum operator for f \in Bp. We note that if
1 < p < \infty , then the operator Tn is bounded [5] on Bp. Hence there exists a constant c > 0 such
that
\| Tn (f, \cdot )\| p \leq c \| f\| p (11)
holds for f \in Bp.
Let \alpha r
n := 1/ (n (n - 1) (n - 2) (n - r + 1)).
Theorem 6. Let n, r \in \BbbN , 1 \leq p \leq \infty and f \in Bp
r . Then
\| f - Tn (f)\| p \leq \alpha r
n
\bigm\| \bigm\| \bigm\| f (r) - Tn - r - 1
\Bigl(
f (r)
\Bigr) \bigm\| \bigm\| \bigm\|
p
holds for r < n.
Proof. Let \rho \in [0, 1) and f\rho (z) := f (\rho z) be dilation operator. Since f \in Bp, 1 \leq p \leq \infty , then
f\rho \in Hp. Hence [20, p. 158]
f\rho
\Bigl(
ei\theta
\Bigr)
- Tn
\Bigl(
f\rho , e
i\theta
\Bigr)
=
eir\theta
2\pi
2\pi \int
0
\Bigl(
f (r)
\rho
\Bigl(
ei(\theta +t)
\Bigr)
- Tn - r - 1
\Bigl(
f (r)
\rho , ei(\theta +t)
\Bigr) \Bigr) \varphi (t)
ei(n - r)t
dt,
where \varphi (t) = \alpha r
n + 2
\sum \infty
k=1 \alpha
r
k+n \mathrm{c}\mathrm{o}\mathrm{s} kt \geq 0. Using generalized Minkowski inequality we get
Mp
p (\rho , f - Tn (f)) =
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| f\rho \Bigl( ei\theta \Bigr) - Tn
\Bigl(
f\rho , e
i\theta
\Bigr) \bigm| \bigm| \bigm| p d\theta =
=
1
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| e
ir\theta
2\pi
2\pi \int
0
\Bigl(
f (r)
\rho
\Bigl(
ei(\theta +t)
\Bigr)
- Tn - r - 1
\Bigl(
f (r)
\rho , ei(\theta +t)
\Bigr) \Bigr)
e - i(n - r)t\varphi (t) dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p
d\theta \leq
\leq
Mp
p
\bigl(
\rho , f (r) - Tn - r - 1
\bigl(
f (r)
\bigr) \bigr)
2\pi
2\pi \int
0
\bigm| \bigm| \bigm| e - i(n - r)t\varphi (t)
\bigm| \bigm| \bigm| dt \leq
\leq \alpha r
nM
p
p
\Bigl(
\rho , f (r) - Tn - r - 1
\Bigl(
f (r)
\Bigr) \Bigr)
and hence the required result
\| f - Tn (f)\| p \leq \alpha r
n
\bigm\| \bigm\| \bigm\| f (r) - Tn - r - 1
\Bigl(
f (r)
\Bigr) \bigm\| \bigm\| \bigm\|
p
follows.
Theorem 6 is proved.
From Theorem 6 and (11) we have the following corollary.
Corollary 3. Let n, r \in \BbbN , 1 < p < \infty and f \in Bp
r . Then
En (f)p \leq cn - rEn - r
\Bigl(
f (r)
\Bigr)
p
holds for r < n.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
444 R. AKGÜN
Bernstein inequality for Bp, 0 < p \leq \infty .
Lemma 2. Let n \in \BbbN and Pn \in \scrP n and 0 < p \leq \infty . Then there exists a constant c > 0,
depending only on p, such that \bigm\| \bigm\| P \prime
n
\bigm\| \bigm\|
p
\leq cn \| Pn\| p .
When p \geq 1 the constant c can be chosen to be 4.
This lemma was proved for p \geq 1 in [23], for p > 0 in [21, p. 432] (the main theorem).
Theorem 7. Let n, r \in \BbbN , 1 < p < \infty , f \in Bp
r , P \ast
n \in \scrP n and En (f)p = \| f - P \ast
n\| p. Then for
all k = 0, r \bigm\| \bigm\| \bigm\| f (k) - P \ast (k)
n
\bigm\| \bigm\| \bigm\|
p
\leq cnk - rEn - r
\Bigl(
f (r)
\Bigr)
p
holds for r \leq n.
Proof. Let qn - k \in \scrP n - k, En - k
\bigl(
f (k)
\bigr)
p
=
\bigm\| \bigm\| f (k) - qn - k
\bigm\| \bigm\|
p
. Then using Corollary 3\bigm\| \bigm\| \bigm\| f (k) - P \ast (k)
n
\bigm\| \bigm\| \bigm\|
p
\leq
\bigm\| \bigm\| \bigm\| f (k) - Tn - k
\Bigl(
f (k)
\Bigr) \bigm\| \bigm\| \bigm\|
p
+
\bigm\| \bigm\| \bigm\| T(k)
n (f) - P \ast (k)
n
\bigm\| \bigm\| \bigm\|
p
\leq
\leq
\bigm\| \bigm\| \bigm\| f (k) - qn - k
\bigm\| \bigm\| \bigm\|
p
+
\bigm\| \bigm\| \bigm\| qn - k - Tn - k
\Bigl(
f (k)
\Bigr) \bigm\| \bigm\| \bigm\|
p
+
\bigm\| \bigm\| \bigm\| (Tn (f) - P \ast
n)
(k)
\bigm\| \bigm\| \bigm\|
p
\leq
\leq En - k
\Bigl(
f (k)
\Bigr)
p
+
\bigm\| \bigm\| \bigm\| Tn - k (qn - k) - Tn - k
\Bigl(
f (k)
\Bigr) \bigm\| \bigm\| \bigm\|
p
+ cnk \| Tn (f) - P \ast
n\| p \leq
\leq (1 + c)En - k
\Bigl(
f (k)
\Bigr)
p
+ cnk \| Tn (f) - Tn (P
\ast
n)\| p \leq
\leq cnk - rEn - r
\Bigl(
f (r)
\Bigr)
p
+ cnkEn (f)p \leq cnk - rEn - r
\Bigl(
f (r)
\Bigr)
p
and the theorem is proved.
Theorem 8. Let n, r \in \BbbN , 1 < p < \infty and f \in Bp
r . Then there exists a \Phi n \in \scrP n such that for
all k = 0, r \bigm\| \bigm\| \bigm\| f (k) - \Phi n
(k)
\bigm\| \bigm\| \bigm\|
p
\leq cnk - r\~\omega n - r
\Bigl(
1/n, f (r)
\Bigr)
p
holds for r \leq n.
Proof. Let P \ast
n \in \scrP n, En (f)p = \| f - P \ast
n\| p and \Phi n = \mathrm{P}n. From (9) and Theorem 5 we have
\| f - \mathrm{P}n\| p \leq cn - r\~\omega n - r
\Bigl(
1/n, f (r)
\Bigr)
p
and \| f - P \ast
n\| p \leq cn - r\~\omega n - r
\Bigl(
1/n, f (r)
\Bigr)
p
.
On the other hand we get
\| \mathrm{P}n - P \ast
n\| p \leq 2cn - r\~\omega n - r
\Bigl(
1/n, f (r)
\Bigr)
p
.
Hence by Bernstein inequality in Lemma 2 we obtain\bigm\| \bigm\| \bigm\| f (k) - \mathrm{P}(k)
n
\bigm\| \bigm\| \bigm\|
p
\leq
\bigm\| \bigm\| \bigm\| f (k) - P \ast (k)
n
\bigm\| \bigm\| \bigm\|
p
+
\bigm\| \bigm\| \bigm\| \mathrm{P}(k)
n - P \ast (k)
n
\bigm\| \bigm\| \bigm\|
p
\leq
\leq cnk - rEn - r
\Bigl(
f (r)
\Bigr)
p
+ cnk - r\~\omega n - r
\Bigl(
1/n, f (r)
\Bigr)
p
\leq cnk - r\~\omega n - r
\Bigl(
1/n, f (r)
\Bigr)
p
.
Theorem 8 is proved.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES 445
5. Converse theorems. Bernstein inequalities are play the central role in the proof of converse
approximation theorems in Approximation Theory. For the spaces Bp these inequalities was estab-
lished by [23] (1 \leq p) and [21] (p > 0). But in some cases Bernstein inequalities are improved
to Nikolskii – Stechkin inequalities. In the present section we prove a Nikolskii – Stechkin-type in-
equality in Bp, 0 \leq p \leq \infty . For this purpose we use a Szego composition theorem for polynomials
in Hp, 0 \leq p \leq \infty . Let Pn,\Lambda n \in \scrP n and we write them of the form Pn (z) =
\sum n
k=0C
k
nckz
k,
\Lambda n (z) =
\sum n
k=0C
k
n\lambda kz
k. Here Ck
n are the Binom coefficients. The Szego composition of Pn and \Lambda n
will be defined as the polynomial Pn \otimes \Lambda n :=
\sum n
k=0C
k
nck\lambda kz
k.
Theorem 9 [2]. Let n \in \BbbN , Pn,\Lambda n \in \scrP n and Pn \otimes \Lambda n be the Szego composition of Pn and \Lambda n.
Then
\| Pn \otimes \Lambda n\| Hp \leq \| \Lambda n\| H0 \| Pn\| Hp , 0 \leq p \leq \infty .
We prove a Nikolskii – Stechkin-type inequality.
Theorem 10. Let n,m \in \BbbN , n > m, Pn \in \scrP n and 0 < t < (2\pi /n). Then
\bigm\| \bigm\| \bigm\| P (m)
n (z)
\bigm\| \bigm\| \bigm\|
p
\leq
m\prod
j=1
(n - m+ j) 2(n - m+j - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ j) t/2
\| [z, Pn]
m
eit\| p , 0 \leq p \leq \infty .
Proof. Let m = 1 and q = eit. Since the polynomials P \prime
n (z) and [z, Pn]
1
q are independent of the
constant term of Pn (z) , we may take Pn (z) of the form
Pn (z) = z
n - 1\sum
k=0
Ck
n - 1akz
k.
Then
P \prime
n (z) =
n - 1\sum
k=0
Ck
n - 1ak (k + 1) zk and [z, Pn]
1
q = z
n - 1\sum
k=0
Ck
n - 1ak
\Bigl(
1 - ei(k+1)t
\Bigr)
zk.
Now taking
Qn - 1 (z) =
n - 1\sum
k=0
Ck
n - 1
k + 1
1 - ei(k+1)t
zk
we get
P \prime
n (z) =
1
z
[z, Pn]
1
q \otimes Qn - 1 (z) .
Using Theorem 9 \bigm\| \bigm\| P \prime
n (z)
\bigm\| \bigm\|
Hp \leq \| Qn - 1 (z)\| H0
\bigm\| \bigm\| \bigm\| [z, Pn]
1
q
\bigm\| \bigm\| \bigm\|
Hp
.
From [16] (inequalities (8) and (10)) we have
\| Qn - 1 (z)\| H0 \leq n2(n - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n}nt/2
and hence \bigm\| \bigm\| P \prime
n (z)
\bigm\| \bigm\|
Hp \leq n2(n - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n}nt/2
\bigm\| \bigm\| \bigm\| [z, Pn]
1
q
\bigm\| \bigm\| \bigm\|
Hp
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
446 R. AKGÜN
By the dilation function method (see, for example, [26]) we conclude for m = 1\bigm\| \bigm\| P \prime
n
\bigm\| \bigm\|
p
\leq n2(n - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n}nt/2
\bigm\| \bigm\| \bigm\| [\cdot , Pn]
1
q
\bigm\| \bigm\| \bigm\|
p
, 0 \leq p \leq \infty . (12)
For m > 1 using (12) and inequalities in (6) we find\bigm\| \bigm\| \bigm\| P (m)
n
\bigm\| \bigm\| \bigm\|
p
\leq (n - m+ 1) 2(n - m)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ 1) t/2
\bigm\| \bigm\| \bigm\| \bigm\| \Bigl[ \cdot , P (m - 1)
n
\Bigr] 1
q
\bigm\| \bigm\| \bigm\| \bigm\|
p
=
=
(n - m+ 1) 2(n - m)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ 1) t/2
\bigm\| \bigm\| \bigm\| \bigm\| d
dz
\biggl( \Bigl[
\cdot , P (m - 2)
n
\Bigr] 1,1
q
\biggr) \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq
2\prod
j=1
(n - m+ j) 2(n - m+j - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ j) t/2
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\biggl[
\cdot ,
\Bigl[
\cdot , P (m - 2)
n
\Bigr] 1,1
q
\biggr] 1
q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
=
2\prod
j=1
(n - m+ j) 2(n - m+j - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ j) t/2
\bigm\| \bigm\| \bigm\| \bigm\| \Bigl[ \cdot , P (m - 2)
n
\Bigr] 2
q
\bigm\| \bigm\| \bigm\| \bigm\|
p
\leq . . .
. . . \leq
m\prod
j=1
(n - m+ j) 2(n - m+j - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ j) t/2
\bigm\| \bigm\| \bigm\| [\cdot , Pn]
m
q
\bigm\| \bigm\| \bigm\|
p
.
Theorem 10 is proved.
As a corollary we get the following Bernstein-type inequality:
Corollary 4. Let n,m \in \BbbN , Pn \in \scrP n, 0 < t < (2\pi /n) and n > m. Then\bigm\| \bigm\| \bigm\| P (m)
n (z)
\bigm\| \bigm\| \bigm\|
p
\leq c
m\prod
j=1
(n - m+ j) 2(n - m+j - 1)/2
\mathrm{s}\mathrm{i}\mathrm{n} (n - m+ j) t/2
\| Pn\| p
holds for 0 \leq p \leq \infty .
From Theorem 5.1 of [4, p. 216, 217] and Lemma 2 we have the following corollary.
Corollary 5. Let r, n \in \BbbN , 0 < p \leq \infty and f \in Bp. Then there exists a constant c > 0
independent of n such that
Kr (f, 1/n) \leq
c
nr
\Biggl\{
n\sum
k=1
\Bigl[
krEk (f)p
\Bigr] \mu 1
k
\Biggr\} 1/\mu
hold where \mu := \mathrm{m}\mathrm{i}\mathrm{n} \{ 1, p\} and Kr (f, \delta ) := \mathrm{i}\mathrm{n}\mathrm{f}
g\in Bp
r
\Bigl\{
\| f - g\| p + \delta r
\bigm\| \bigm\| g(r)\bigm\| \bigm\|
p
\Bigr\}
, \delta \geq 0 is the Peetre
K-functional.
Theorem 11. Let m \in \BbbN , 0 < p \leq \infty and f \in Bp. Then there exists a constant c > 0
independent of n such that
\~\omega m (1/n, f)p \leq
c
nr
\Biggl\{
n\sum
k=1
\Bigl[
krEk (f)p
\Bigr] \mu 1
k
\Biggr\} 1/\mu
hold where \mu := \mathrm{m}\mathrm{i}\mathrm{n} \{ 1, p\} .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
POLYNOMIAL APPROXIMATION IN BERGMAN SPACES 447
Proof. Using Corollary 1 we have
\~\omega \mu
m (1/n, f)p \leq c
\biggl(
\| f - g\| \mu p + (1/n)\mu m
\bigm\| \bigm\| \bigm\| g(m)
\bigm\| \bigm\| \bigm\| \mu
p
\biggr)
\leq cK\mu
m (1/n, f)p
and the estimate \~\omega m (1/n, f)p \leq cKm (1/n, f)p fulfilled. By Corollary 5 we conclude the required
result.
Theorem 11 is proved.
This theorem is a generalization of converse theorem of [13] (0 < p < 1).
As an application of Theorems 11 and 4 we get the Marchaud inequality for 0 < p < 1.
Corollary 6. Let r,m \in \BbbN , r > m, 0 < p < 1 and f \in Bp. Then for all 0 < \delta < 1/r
\~\omega m (\delta , f)p \leq c\delta m
\left\{
1/r\int
\delta
\~\omega r (t, f)
p
p
tmp+1
dt
\right\}
1/p
holds.
Definition 1. Let \varphi (t) be a positive function for t > 0 and \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0 \varphi (t) = 0. We suppose that
0 < p \leq \infty , m \in \BbbN ,
H\varphi
p,m :=
\Bigl\{
f \in Bp : \~\omega m (t, f)p = \scrO (\varphi (t)) , t \rightarrow 0
\Bigr\}
and \mathrm{L}\mathrm{i}\mathrm{p} (\alpha , p) \equiv Ht\alpha
p,m, \alpha > 0.
Corollary 7. Let n \in \BbbN and f \in Bp. In this case
(A) Suppose that 0 < p < 1. Then:
(i) for 0 < \alpha < 1 the conditions En (f)p = O (n - \alpha ) and f \in \mathrm{L}\mathrm{i}\mathrm{p} (\alpha , p) are equivalent;
(ii) for \alpha = 1 the condition En (f)p = O
\bigl(
n - 1
\bigr)
implies \~\omega m (\delta , f)p = O
\Bigl(
\delta | \mathrm{l}\mathrm{n} \delta | 1/p
\Bigr)
;
(iii) for \alpha > 1 the condition En (f)p = O (n - \alpha ) implies f \in \mathrm{L}\mathrm{i}\mathrm{p} (1, p) .
(B) Suppose that 1 \leq p \leq \infty . Then:
(i) for 0 < \alpha < 1 the conditions En (f)p = O (n - \alpha ) and f \in \mathrm{L}\mathrm{i}\mathrm{p} (\alpha , p) are equivalent;
(ii) for \alpha = 1 the condition En (f)p = O
\bigl(
n - 1
\bigr)
implies \~\omega m (\delta , f)p = O (\delta | \mathrm{l}\mathrm{n} \delta | ) .
Corollary 8. Let n,m \in \BbbN , 0 < p < 1 and f \in Bp. If
1\int
\delta
\varphi (t)p
tmp+1
dt = O
\biggl(
\varphi (\delta )p
\delta mp
\biggr)
, \delta \rightarrow 0,
then, the conditions f \in H\varphi
p,m and E2n (f)p = O (\varphi (2 - n)) , n \rightarrow \infty , are equivalent.
6. Concluding remarks. Theorems 7 and 8 for 0 < p < 1 and for p = \infty , remain open. It may
be interesting to investigate above problems in B0 or Bp (\Omega ) for general domains \Omega .
References
1. Al’per S. Ja. Approximation of analytic functions in the mean over a region (in Russian) // Dokl. Akad. Nauk SSSR. –
1961. – 136. – P. 265 – 268.
2. Arestov V. V. Integral inequalities for trigonometric polynomials and their derivatives // Izv. Akad. Nauk SSSR.
Ser. Mat. – 1981. – 45, № 1. – P. 3 – 22.
3. Burbea J. Polynomial approximation in Bers spaces of Carathéodory domains // J. London Math. Soc. (2). – 1977. –
15, № 2. – P. 255 – 266.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
448 R. AKGÜN
4. DeVore R. A., Lorentz G. G. Constructive approximation // Fund. Princip. Math. Sci. – Berlin: Springer-Verlag,
1993. – 303.
5. Duren P., Schuster A. Bergman spaces // Math. Surv. and Monogr. – Providence, RI: Amer. Math. Soc., 2004. – 100.
6. Gaier D. Lectures on complex approximation. – 1987.
7. Kryakin Yu. V. Jackson theorems in Hp; 0 < p < \infty (in Russian) // Izv. Vysch. Uchebn. Zaved. Mat. – 1985. –
№ 8. – P. 19 – 23.
8. Kryakin Yu., Trebels W. q-Moduli of continuity in Hp(D), p > 0, and an inequality of Hardy and Littlewood //
J. Approxim. Theory. – 2002. – 115, № 2, – P. 238 – 259.
9. Metzger T. A. On polynomial approximation in Aq(D) // Proc. Amer. Math. Soc. – 1973. – 37. – P. 468 – 470.
10. Quade E. S. Trigonometric approximation in the mean // Duke Math. J. – 1937. – 3, № 3. – P. 529 – 543.
11. Ren Guangbin, Wang Mingzhi. Holomorphic Jackson’s theorems in polydiscs // J. Approxim. Theory. – 2005. – 134,
№ 2. – P. 175 – 198.
12. Shabozov M. Sh., Shabozov O. Sh. Best approximation and the value of the widths of some classes of functions in
the Bergman space Bp, 1 \leq p \leq \infty (in Russian) // Dokl. Akad. Nauk. – 2006. – 410, № 4. – P. 461 – 464.
13. Shen Xie Chang, Xing Fu Chong. The problem of best approximation by polynomials in the spaces Hp
q (0 < p < 1,
q > 1) (in Chinese) // J. Math. Res. Exposit. – 1989. – 9, № 1. – P. 107 – 114.
14. Storozhenko E. A. Theorems of Jackson-type in Hp, 0 < p < 1 (in Russian) // Izv. Akad. Nauk SSSR. Ser. Mat. –
1980. – 44, № 4. – P. 946 – 962.
15. Storozhenko E. A. On a Hardy – Littlewood problem (in Russian) // Mat. Sb. (N. S.). – 1982. – 119(161), № 4. –
P. 564 – 583.
16. Storozhenko E. A. The Nikol’skĭı – Stechkin inequality for trigonometric polynomials in L0 // Mat. Zametki. – 2006. –
80, № 3. – P. 421 – 428.
17. Suetin P. K. Series of Faber polynomials. – Reading, 1994.
18. Vakarchuk S. B. On the best polynomial approximation in some Banach spaces for functions analytical in the unit
disc // Math. Notes. – 1994. – 55, № 4. – P. 338 – 343.
19. Wang Ming Zhi, Ren Guang Bin. Jackson’s theorem on bounded symmetric domains // Acta Math. sinica (Engl. Ser.). –
2007. – 23, № 8. – P. 1391 – 1404.
20. Taikov L. V. Best approximation in the mean of certain classes of analytic functions (in Russian) // Mat. Zametki. –
1967. – 1, № 2. – P. 155 – 162.
21. Xing Fu Chong. A Bernstein-type inequality in Bergman spaces Bp
q (p > 0, q > 1) (in Chinese) // Acta Math. sinica
(Chin. Ser.). – 2006. – 49, № 2. – P. 431 – 434.
22. Xing Fu Chong. An inverse theorem for the best approximation by polynomials in Bergman spaces Bp
q (p > 0, q > 1)
(in Chinese) // Acta Math. sinica (Chin. Ser.). – 2006. – 49, № 3. – P. 519 – 524.
23. Xing Fu Chong, Su Chao Long. On some properties of Hp
q space // J. Xinjiang Univ. Nat. Sci. – 1986. – 3, № 3. –
P. 49 – 56.
24. Xing Fu Chong, Su Zhalong. Inverse theorems to the best approximation by polynomials in Hp
q (p \geq 1, q > 1)
spaces // J. Xinjiang Univ. Natur. Sci. – 1989. – 6, № 4. – P. 36 – 46.
25. Zhong Le Fan. Interpolation polynomial approximation in Bergman spaces (in Chinese) // Acta Math. sinica. – 1991. –
34, № 1. – P. 56 – 66.
26. Zhu Kehe. Translating inequalities between Hardy and Bergman spaces // Amer. Math. Mon. – 2004. – 111, № 6. –
P. 520 – 525.
Received 11.10.12,
after revision — 24.11.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
|
| id | umjimathkievua-article-1850 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:53Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ea/f15dd7bf4cc0c396380d4e725e2f3aea.pdf |
| spelling | umjimathkievua-article-18502019-12-05T09:29:54Z Polynomial approximation in Bergman spaces Полiномiальнi наближення у просторах Бергмана Akgün, R. Акгюн, Р. The purpose of this work is to obtain Jackson and converse inequalities of the polynomial approximation in Bergman spaces. Some known results presented for the moduli of continuity are extended to the moduli of smoothness. We proved some simultaneous approximation theorems and obtained the Nikolskii – Stechkin inequality for polynomials in these spaces. Meтою даної роботи є встановлення нерiвностi Джексона та обернених нерiвностей для полiномiальних наближень у просторi Бергмана. Деякi вiдомi результати для модулiв неперервностi узагальнено на модулi гладкостi. Доведено деякi спiльнi теореми про наближення та встановлено нерiвнiсть Нiкольського – Стечкiна для полiномiв у цих просторах. Institute of Mathematics, NAS of Ukraine 2016-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1850 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 4 (2016); 435-448 Український математичний журнал; Том 68 № 4 (2016); 435-448 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1850/832 Copyright (c) 2016 Akgün R. |
| spellingShingle | Akgün, R. Акгюн, Р. Polynomial approximation in Bergman spaces |
| title | Polynomial approximation in Bergman spaces |
| title_alt | Полiномiальнi наближення у просторах Бергмана |
| title_full | Polynomial approximation in Bergman spaces |
| title_fullStr | Polynomial approximation in Bergman spaces |
| title_full_unstemmed | Polynomial approximation in Bergman spaces |
| title_short | Polynomial approximation in Bergman spaces |
| title_sort | polynomial approximation in bergman spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1850 |
| work_keys_str_mv | AT akgunr polynomialapproximationinbergmanspaces AT akgûnr polynomialapproximationinbergmanspaces AT akgunr polinomialʹninabližennâuprostorahbergmana AT akgûnr polinomialʹninabližennâuprostorahbergmana |