Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems
UDC 517.5 We prove some new strong convergence theorems for partial sums and Fejér means with respect to the Vilenkin system.  
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| author | Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Tutberidze, Giorgi |
| author_facet | Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Tutberidze, Giorgi |
| author_sort | Persson, L.-E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:15Z |
| description | UDC 517.5
We prove some new strong convergence theorems for partial sums and Fejér means with respect to the Vilenkin system.
  |
| doi_str_mv | 10.37863/umzh.v73i4.226 |
| first_indexed | 2026-03-24T02:02:18Z |
| format | Article |
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DOI: 10.37863/umzh.v73i4.226
UDC 517.5
L.-E. Persson (UiT -The Arctic Univ. Narvik, Norway and Karlstad Univ., Sweden),
G. Tephnadze (Univ. Georgia, School Sci. and Technology, Tbilisi, Georgia),
G. Tutberidze (Univ. Georgia, School Sci. and Technology, Georgia and UiT - The Arctic Univ. Narvik, Norway),
P. Wall (Luleå Univ. Technology, Sweden)
SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR
MEANS WITH RESPECT TO VILENKIN SYSTEMS*
ДЕЯКI НОВI РЕЗУЛЬТАТИ ЩОДО СТРОГОЇ ЗБIЖНОСТI СЕРЕДНIХ ФЕЄРА
ВIДНОСНО СИСТЕМ ВIЛЕНКIНА
We prove some new strong convergence theorems for partial sums and Fejér means with respect to the Vilenkin system.
Доведено деякi теореми про строгу збiжнiсть часткових сум та середнiх Феєра вiдносно системи Вiленкiна.
1. Introduction. Concerning definitions and notations used in this introductions we refer to Secti-
on 2.
It is well-known (for details see, e.g., [1, 8, 10]) that Vilenkin system forms not basis in the space
L1(Gm). Moreover, there is a function in the martingale Hardy space H1(Gm) such that the partial
sums of f are not bounded in L1(Gm)-norm. However, for all p > 0 and f \in Hp, there exists an
absolute constant cp such that
\| SMk
f\| p \leq cp\| f\| Hp . (1)
In [5] (see also [11]) the following strong convergence result was obtained for all f \in H1(Gm):
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\| Skf - f\| 1
k
= 0.
It follow that
1
\mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\| Skf\| 1
k
\leq \| f\| H1 , n = 2, 3, . . . .
In [19] was proved that for any f \in H1 there exists an absolute constant c such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
1
n \mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\| Skf\| 1 \leq \| f\| H1 , n = 1, 2, 3, . . . .
Moreover, for every nondecreasing function \varphi : \BbbN + \rightarrow [1,\infty ), satisfying the condition
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{o}\mathrm{g} n
\varphi n
= +\infty ,
there exists a function f \in \BbbN 1 such that
* The research of third author is supported by Shota Rustaveli National Science Foundation (grant no. PHDF-18-476).
c\bigcirc L.-E. PERSSON, G. TEPHNADZE, G. TUTBERIDZE, P. WALL, 2021
544 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR MEANS . . . 545
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
1
n\varphi n
n\sum
k=1
\| Skf\| 1 = \infty .
For the Vilenkin system Simon [12] proved that there is an absolute constant cp, depending only
on p, such that
\infty \sum
k=1
\| Skf\| pp
k2 - p
\leq cp\| f\| pHp
for all f \in Hp(Gm), where 0 < p < 1. In [16] was proved that for any nondecreasing function \Phi :
\BbbN + \rightarrow [1,\infty ), satisfying the condition \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \Phi (n) = +\infty , there exists a martingale f \in Hp(Gm)
such that
\infty \sum
k=1
\| Skf\| pweak-Lp
\Phi (k)
k2 - p
= \infty for 0 < p < 1.
Strong convergence theorems of two-dimensional partial sums was investigate by Weisz [23],
Goginava [6], Gogoladze [7], Tephnadze [18] (see also [9]).
Weisz [24] considered the norm convergence of Fejér means of Walsh – Fourier series and proved
the following theorem.
Theorem W1 (Weisz). Let p > 1/2 and f \in Hp. Then
\| \sigma kf\| p \leq cp\| f\| Hp .
Moreover, Weisz [24] also proved that, for all p > 0 and f \in Hp, there exists an absolute
constant cp such that
\| \sigma Mk
f\| p \leq cp\| f\| Hp . (2)
Theorem W1 implies that
1
n2p - 1
n\sum
k=1
\| \sigma kf\| pp
k2 - 2p
\leq cp\| f\| pHp
, 1/2 < p <\infty .
If Theorem W1 should hold for 0 < p \leq 1
2
, then we have
\infty \sum
k=1
\| \sigma kf\| pp
k2 - 2p
\leq cp\| f\| pHp
, 0 < p < 1/2, (3)
1
\mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\| \sigma kf\|
1/2
1/2
k
\leq c\| f\| 1/2H1/2
, (4)
and
1
n
n\sum
k=1
\| \sigma kf\|
1/2
1/2 \leq c\| f\| 1/2H1/2
. (5)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
546 L.-E. PERSSON, G. TEPHNADZE, G. TUTBERIDZE, P. WALL
However, in [14] (see also [2, 3]) it was proved that the assumption p > 1/2 in Theorem W1 is
essential. In particular, there exists a martingale f \in H1/2 such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
\| \sigma nf\| 1/2 = +\infty .
In [4] (see also [17]) it was proved that (3) and (4) hold though Theorem W1 is not true for
0 < p \leq 1/2.
Moreover, in [4] it was proved that if 0 < p < 1/2 and \Phi : \BbbN + \rightarrow [1,\infty ) be any nondecreasing
function satisfying condition
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
k2 - 2p
\Phi k
= \infty ,
then there exists a martingale f \in Hp such that
\infty \sum
m=1
\| \sigma mf\| pweak-Lp
\Phi m
= \infty .
On the other hand, for the Walsh system (5) does not hold (see [17]). In particular, it was proved
that there exists a martingale f \in H1/2 such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
1
n
n\sum
m=1
\| \sigma mf\| 1/21/2 = \infty . (6)
In this paper, we prove more general result for bounded Vilenkin system. In special case we also
obtain (6).
This paper is organized as follows. In order not to disturb our discussions later on some definitions
and notations are presented in Section 2. For the proofs of the main results we need some auxiliary
lemmas, some of them are new and of independent interest. These results are presented in Section 3.
The main result with proof is given in Section 4.
2. Definitions and notations. Let \BbbN + denote the set of the positive integers, \BbbN := \BbbN + \cup \{ 0\} .
Let m := (m0,m1, . . .) denote a sequence of the positive integers not less than 2.
Denote by
Zmk
:= \{ 0, 1, . . . ,mk - 1\}
the additive group of integers modulo mk.
Define the group Gm as the complete direct product of the group Zmj with the product of the
discrete topologies of Zmj .
The direct product \mu of the measures
\mu k(\{ j\} ) := 1/mk, j \in Zmk
,
is the Haar measure on Gm with \mu (Gm) = 1.
If \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN mn < \infty , then we call Gm a bounded Vilenkin group. If the generating sequence
m is not bounded, then Gm is said to be an unbounded Vilenkin group. In this paper, we discuss
bounded Vilenkin groups only.
The elements of Gm are represented by sequences
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR MEANS . . . 547
x := (x0, x1, . . . , xk, . . .), xk \in Zmk
.
It is easy to give a base for the neighbourhood of Gm namely
I0(x) := Gm,
and
In(x) :=
\bigl\{
y \in Gm | y0 = x0, . . . , yn - 1 = xn - 1
\bigr\}
, x \in Gm, n \in \BbbN .
Denote In := In(0) for n \in \BbbN and In := Gm\setminus In.
Let
en := (0, . . . , 0, xn = 1, 0, . . .) \in Gm, n \in \BbbN .
If we define the so-called generalized number system based on m in the following way:
M0 := 1, Mk+1 := mkMk, k \in \BbbN ,
then every n \in \BbbN can be uniquely expressed as n =
\sum \infty
k=0
njMj , where nj \in Zmj , j \in \BbbN , and
only a finite number of nj `s differ from zero. Let \| n\| := \mathrm{m}\mathrm{a}\mathrm{x}\{ j \in \BbbN , nj \not = 0\} .
For the natural number n =
\sum \infty
j=1
njMj , we define
\delta j = \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}nj = \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n} (\ominus nj), \delta \ast j = \| \ominus nj - 1\| \delta j ,
where \ominus is the inverse operation for ak \oplus bk = (ak + bk)modmk.
We define functions v and v\ast by
v(n) =
\infty \sum
j=0
| \delta j+1 - \delta j | + \delta 0, v\ast (n) =
\infty \sum
j=0
\delta \ast j .
The nth Lebesgue constant is defined in the following way:
Ln = \| Dn\| 1.
The norm (or quasinorm) of the space Lp(Gm) is defined by
\| f\| p :=
\left( \int
Gm
\| f(x)\| pd\mu (x)
\right) 1/p
, 0 < p <\infty .
The space weak-Lp(Gm) consists of all measurable functions f for which
\| f\| weak-Lp(Gm) := \mathrm{s}\mathrm{u}\mathrm{p}
\lambda >0
\lambda p\mu \{ f > \lambda \} < +\infty .
Next, we introduce on Gm an orthonormal system which is called the Vilenkin system.
At first, define the complex valued function rk(x) : Gm \rightarrow \BbbC , the generalized Rademacher
functions as
rk(x) := \mathrm{e}\mathrm{x}\mathrm{p} (2\pi \imath xk/mk) , \imath 2 = - 1, x \in Gm, k \in \BbbN .
Now define the Vilenkin system \psi := (\psi n : n \in \BbbN ) on Gm as
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
548 L.-E. PERSSON, G. TEPHNADZE, G. TUTBERIDZE, P. WALL
\psi n(x) :=
\infty \prod
k=0
rnk
k (x), n \in \BbbN .
Specially, we call this system the Walsh – Paley one if m \equiv 2.
The Vilenkin system is orthonormal and complete in L2(Gm) (for details see, e.g., [1, 10, 20]).
If f \in L1(Gm), then we can define Fourier coefficients, partial sums of the Fourier series, Fejér
means, Dirichlet and Fejér kernels with respect to the Vilenkin system in the usual manner:
\widehat f(k) := \int
Gm
f\psi kd\mu , k \in \BbbN ,
Snf :=
n - 1\sum
k=0
\widehat f(k)\psi k, n \in \BbbN +, S0f := 0,
\sigma nf :=
1
n
n - 1\sum
k=0
Skf, n \in \BbbN +,
Dn :=
n - 1\sum
k=0
\psi k, n \in \BbbN +,
Kn :=
1
n
n - 1\sum
k=0
Dk, n \in \BbbN +.
Recall that (for details see, e.g., [1, 8])
DMn(x) =
\left\{ Mn, x \in In,
0, x /\in In.
(7)
and
DsnMn = DMn
sn - 1\sum
k=0
\psi kMn = DMn
sn - 1\sum
k=0
rkn, 1 \leq sn \leq mn - 1. (8)
The \sigma -algebra generated by the intervals \{ In(x) : x \in Gm\} will be denoted by \digamma n, n \in \BbbN .
Denote by f = (fn, n \in \BbbN ) a martingale with respect to \digamma n, n \in \BbbN (for details see, e.g., [21]). The
maximal function of a martingale f is defined by
f\ast = \mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
| fn| .
In the case f \in L1(Gm), the maximal functions are also be given by
f\ast (x) = \mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
1
| In(x)|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
In(x)
f(u)\mu (u)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR MEANS . . . 549
For 0 < p <\infty the Hardy martingale spaces Hp(Gm) consist of all martingales for which
\| f\| Hp := \| f\ast \| p <\infty .
If f \in L1(Gm), then it is easy to show that the sequence (SMnf : n \in \BbbN ) is a martingale. If
f = (fn, n \in \BbbN ) is martingale, then the Vilenkin – Fourier coefficients must be defined in a slightly
different manner: \widehat f(i) := \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\int
Gm
fk(x)\psi i(x) d\mu (x).
The Vilenkin – Fourier coefficients of f \in L1(Gm) are the same as those of the martingale
(SMnf : n \in \BbbN ) obtained from f.
A bounded measurable function a is p-atom, if there exist an interval I such that\int
I
ad\mu = 0, \| a\| \infty \leq \mu (I) - 1/p, supp (a) \subset I.
3. Auxiliary lemmas.
Lemma 1 [21, 22]. A martingale f = (fn, n \in \BbbN ) is in Hp, 0 < p \leq 1, if and only if there
exist a sequence (ak, k \in \BbbN ) of p-atoms and a sequence (\mu k, k \in \BbbN ) of real numbers such that, for
every n \in \BbbN ,
\infty \sum
k=0
\mu kSMnak = fn a.e., (9)
where
\infty \sum
k=0
| \mu k| p <\infty .
Moreover,
\| f\| Hp \backsim \mathrm{i}\mathrm{n}\mathrm{f}
\Biggl( \infty \sum
k=0
| \mu k| p
\Biggr) 1/p
,
where the infimum is taken over all decomposition of f of the form (9).
By using atomic decomposition of f \in Hp martingales, we can derive a counterexample, which
play a central role to prove sharpness of main results and it will be used several times in the paper.
Lemma 2 [13]. Let n \in \BbbN and 1 \leq sn \leq mn - 1. Then
snMnKsnMn =
sn - 1\sum
l=0
\Biggl(
l - 1\sum
t=0
rtn
\Biggr)
MnDMn +
\Biggl(
sn - 1\sum
l=0
rln
\Biggr)
MnKMn
and
| snMnKsnMn(x)| \geq
M2
n
2\pi
for x \in In+1(en - 1 + en).
Moreover, if x \in It/It+1, x - xtet /\in In and n > t, then
KsnMn(x) = 0. (10)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
550 L.-E. PERSSON, G. TEPHNADZE, G. TUTBERIDZE, P. WALL
Lemma 3 [4]. Let n =
\sum r
i=1
sniMni , where nn1 > nn2 > . . . > nnr \geq 0 and 1 \leq sni < mni
for all 1 \leq i \leq r as well as n(k) = n -
\sum k
i=1
sniMni , where 0 < k \leq r. Then
nKn =
r\sum
k=1
\left( k - 1\prod
j=1
r
snj
nj
\right) snk
Mnk
Ksnk
Mnk
+
r - 1\sum
k=1
\left( k - 1\prod
j=1
r
snj
nj
\right) n(k)Dsnk
Mnk
.
Lemma 4. Let
n =
s\sum
i=1
mi\sum
k=li
nkMk,
where
0 \leq l1 \leq m1 \leq l2 - 2 < l2 \leq m2 \leq . . . \leq ls - 2 < ls \leq ms.
Then
n| Kn(x)| \geq cM2
li
for x \in Ili+1 (eli - 1 + eli) ,
where \lambda = \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN mn and c is an absolute constant.
Proof. Let x \in Ili+1(eli - 1 + eli). By combining (10), (7) and (8), we obtain
Dli = 0
and
Dsnk
Msnk
= Ksnk
Msnk
= 0, snk
> li.
Since sn1 > sn2 > . . . > snr \geq 0, we find
n(k) = n -
k\sum
i=1
sniMni =
s\sum
i=k+1
sniMni \leq
nk+1\sum
i=0
(mi - 1)Mi = mnk+1
Mnk+1
- 1 \leq Mnk
.
According to Lemma 3, we have
n| Kn| \geq | sliMliKsliMli
| -
i - 1\sum
r=1
mr\sum
k=lr
| skMkKskMk
| -
i - 1\sum
r=1
mr\sum
k=lr
| MkDskMk
| = I1 - I2 - I3.
Let x \in Ili+1(eli - 1 + eli) and 1 \leq sli \leq mli - 1. By using Lemma 2, we get
I1 = | sliMliKsliMli
| \geq
M2
li
2\pi
\geq
2M2
li
9
.
It is easy to see that
k\sum
s=0
n2sM
2
s \leq
k\sum
s=0
(ms - 1)2M2
s \leq
k\sum
s=0
m2
sM
2
s - 2
k\sum
s=0
msM
2
s +
k\sum
s=0
M2
s =
=
k\sum
s=0
M2
s+1 - 2
k\sum
s=0
Ms+1Ms +
k\sum
s=0
M2
s =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR MEANS . . . 551
=M2
k+1 + 2
k\sum
s=0
M2
s - 2
k\sum
s=0
Ms+1Ms - M2
0 \leq M2
k+1 - 1
and
k\sum
s=0
nsMs \leq
k\sum
s=0
(ms - 1)Ms = mkMk - m0M0 \leq Mk+1 - 2.
Since mi - 1 \leq li - 2 if we use the estimates above, then we obtain
I2 \leq
li - 2\sum
s=0
| nsMsKnsMs(x)| \leq
li - 2\sum
s=0
nsMs
nsMs + 1
2
\leq
\leq (mli - 2 - 1)Mli - 2
2
li - 2\sum
s=0
(nsMs + 1) \leq
\leq (mli - 2 - 1)Mli - 2
2
Mli - 1 +
(mli - 2 - 1)Mli - 2
2
li \leq
\leq
M2
li - 1
2
- Mli - 2Mli - 1
2
+Mli - 1li. (11)
For I3 we have
I3 \leq
li - 2\sum
k=0
| MkDnkMk
(x)| \leq
li - 2\sum
k=0
nkM
2
k \leq Mli - 2
li - 2\sum
k=0
nkMk \leq Mli - 1Mli - 2 - 2Mli - 2. (12)
By combining (11), (12), we get
n| Kn(x)| \geq I1 - I2 - I3 \geq
M2
li
2\pi
+
3
2
+ 2Mli - 2 -
Mli - 1Mli - 2
2
-
M2
li - 1
2
- Mli - 1li \geq
\geq
M2
li
2\pi
-
M2
li
16
-
M2
li
8
+
7
2
- Mli - 1li \geq
\geq
2M2
li
9
-
3M2
li
16
+
7
2
- Mli - 1li \geq
M2
li
144
- Mli - 1li.
Suppose that li \geq 4. Then
n| Kn(x)| \geq
M2
li
36
- Mli
4
\geq
M2
li
36
-
M2
li
64
\geq
5M2
li
36 \cdot 16
\geq
M2
li
144
.
Lemma 4 is proved.
4. Main result. The main result of this paper is the following theorem.
Theorem 1. 1. Let f \in H1/2. Then there exists an absolute constant c such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN
1
n \mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\| \sigma kf\|
1/2
H1/2
\leq c\| f\| 1/2H1/2
, n = 1, 2, 3, . . . .
2. Let \varphi : \BbbN + \rightarrow [1,\infty ) be a nondecreasing function satisfying the condition
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
552 L.-E. PERSSON, G. TEPHNADZE, G. TUTBERIDZE, P. WALL
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{o}\mathrm{g} n
\varphi n
= +\infty . (13)
Then there exists a function f \in H1/2 such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN +
1
n\varphi n
n\sum
k=1
\| \sigma kf\|
1/2
H1/2
= \infty .
Corollary 1. There exists a martingale f \in H1/2 such that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN +
1
n
n\sum
k=1
\| \sigma kf\|
1/2
1/2 = \infty .
Proof of Theorem 1. 1. In [15] was proved that there exists an absolute constant c, such that
\| \sigma kf\|
1/2
H1/2
\leq c \mathrm{l}\mathrm{o}\mathrm{g} k\| f\| 1/2H1/2
, k = 1, 2, . . . .
Hence,
1
n \mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\| \sigma kf\|
1/2
H1/2
\leq
c\| f\| 1/2H1/2
n \mathrm{l}\mathrm{o}\mathrm{g} n
n\sum
k=1
\mathrm{l}\mathrm{o}\mathrm{g} k \leq c\| f\| 1/2H1/2
.
2. Under the condition (13) there exists an increasing sequence of the positive integers \{ \alpha k :
k \in \BbbN \} such that
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\mathrm{l}\mathrm{o}\mathrm{g}M\alpha k
\varphi 2M\alpha k
= +\infty
and
\infty \sum
k=0
\varphi
1/2
2M\alpha k
\mathrm{l}\mathrm{o}\mathrm{g}1/2M\alpha k
< c <\infty . (14)
Let f = (fn, n \in \BbbN ) be martingale, defined by
fn :=
\sum
\{ k;2\alpha k<n\}
\lambda kak,
where
ak =M\alpha k
r\alpha k
DM\alpha k
=M\alpha k
(D2M\alpha k
- DM\alpha k
)
and
\lambda k =
\varphi 2M\alpha k
\mathrm{l}\mathrm{o}\mathrm{g}M\alpha k
.
Since
S2Aak =
\Biggl\{
ak, \alpha k < A,
0, \alpha k \geq A,
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} (ak) = I\alpha k
,
\int
I\alpha k
akd\mu = 0, \| ak\| \infty \leq M2
\alpha k
= \mu (\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} ak)
- 2,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR MEANS . . . 553
if we apply Lemma 1 and (14), we conclude that f \in H1/2.
Moreover,
\widehat f(j) =
\left\{ M\alpha k
\lambda k, j \in \{ M\alpha k
, . . . , 2M\alpha k
- 1\} , k \in \BbbN ,
0, j /\in
\infty \bigcup
k=1
\{ M\alpha k
, . . . , 2M\alpha k
- 1\} . (15)
We have
\sigma nf =
1
n
M\alpha k
- 1\sum
j=0
Sjf +
1
n
n - 1\sum
j=M\alpha k
Sjf = I + II. (16)
Let M\alpha k
\leq j < 2M\alpha k
. Since
Dj+M\alpha k
= DM\alpha k
+ \psi M\alpha k
Dj , when j \leq M\alpha k
,
if we apply (15), we obtain
Sjf = SM\alpha k
f +
j - 1\sum
v=M\alpha k
\widehat f(v)\psi v = SM\alpha k
f +M\alpha k
\lambda k
j - 1\sum
v=M\alpha k
\psi v =
= SM\alpha k
f +M\alpha k
\lambda k(Dj - DM\alpha k
) = SM\alpha k
f + \lambda k\psi M\alpha k
Dj - M\alpha k
. (17)
According to (17) concerning II, we conclude that
II =
n - M\alpha k
n
SM\alpha k
f +
\lambda kM\alpha k
n
n - 1\sum
j=M2\alpha k
\psi M\alpha k
Dj - M\alpha k
= II1 + II2.
We can estimate II2 as follows:
| II2| =
\lambda kM\alpha k
n
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \psi M\alpha k
n - M\alpha k
- 1\sum
j=0
Dj
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = \lambda kM\alpha k
n
(n - M\alpha k
)| Kn - M\alpha k
| \geq
\geq \lambda k(n - M\alpha k
)| Kn - M\alpha k
| .
Let n =
\sum s
i=1
\sum mi
k=li
Mk, where
0 \leq l1 \leq m1 \leq l2 - 2 < l2 \leq m2 \leq . . . \leq ls - 2 < ls \leq ms.
By applying Lemma 4, we get
| II2| \geq c\lambda k
\bigm| \bigm| \bigm| (n - M\alpha k
)Kn - M\alpha k
(x)
\bigm| \bigm| \bigm| \geq c\lambda kM
2
li
for x \in Ili+1(eli - 1 + eli).
Hence \int
Gm
| II2| 1/2d\mu \geq
s - 1\sum
i=1
\int
Ili+1(eli - 1+eli )
| II2| 1/2d\mu \geq c
s - 1\sum
i=1
\int
Ili+1(eli - 1+eli )
\lambda
1/2
k Mlid\mu \geq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
554 L.-E. PERSSON, G. TEPHNADZE, G. TUTBERIDZE, P. WALL
\geq c\lambda
1/2
k (s - 1) \geq c\lambda
1/2
k v(n - M\alpha k
). (18)
In view of (1), (2) and (16), we find
\| I\| 1/2 =
\bigm\| \bigm\| \bigm\| \bigm\| M\alpha k
n
\sigma M\alpha k
f
\bigm\| \bigm\| \bigm\| \bigm\| 1/2
1/2
\leq
\bigm\| \bigm\| \bigm\| \sigma M\alpha k
f
\bigm\| \bigm\| \bigm\| 1/2
1/2
\leq c \| f\| 1/2H1/2
(19)
and
\| II1\| 1/2 =
\bigm\| \bigm\| \bigm\| \bigm\| n - M\alpha k
n
SM\alpha k
f
\bigm\| \bigm\| \bigm\| \bigm\| 1/2
1/2
\leq
\bigm\| \bigm\| \bigm\| SM\alpha k
f
\bigm\| \bigm\| \bigm\| 1/2
1/2
\leq c \| f\| 1/2H1/2
. (20)
By combining (18) – (20), we have
\| \sigma nf\| 1/21/2 \geq \| II2\| 1/21/2 - \| II1\| 1/21/2 - \| I\| 1/21/2 \geq c\lambda
1/2
k v(n - M\alpha k
) - c\| f\| 1/2H1/2
.
By using estimates with the above, we conclude that
\mathrm{s}\mathrm{u}\mathrm{p}
n\in \BbbN +
1
n\varphi n
n\sum
k=1
\| \sigma kf\|
1/2
1/2 \geq
1
M\alpha k+1\varphi 2M\alpha k
\sum
\{ M\alpha k
\leq l\leq 2M\alpha k\}
\| \sigma lf\|
1/2
1/2 \geq
\geq c
M\alpha k+1\varphi 2M\alpha k
\sum
\{ M\alpha k
\leq l\leq 2M\alpha k\}
\Bigl(
\lambda
1/2
k v (l - M\alpha k
) - c\| f\| 1/2H1/2
\Bigr)
\geq
\geq
c\lambda
1/2
k
M\alpha k
\varphi 2M\alpha k
M\alpha k\sum
l=1
v(l) -
c\| f\| 1/2H1/2
M\alpha k
\varphi 2M\alpha k
\sum
\{ M\alpha k
\leq l\leq 2M\alpha k\}
1 \geq
\geq
c\lambda
1/2
k
M\alpha k
\varphi 2M\alpha k
M\alpha k
- 1\sum
l=1
v(l) - c \geq c
\mathrm{l}\mathrm{o}\mathrm{g}1/2M\alpha k
\varphi
1/2
2M\alpha k
\rightarrow \infty as k \rightarrow \infty .
Theorem 1 is proved.
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SOME NEW RESUTS CONCERING STRONG CONVERGENCE OF FEJÉR MEANS . . . 555
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Received 16.07.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
|
| id | umjimathkievua-article-226 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:18Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d6/584610f3c3a1ea651680dbe9dafc3dd6.pdf |
| spelling | umjimathkievua-article-2262025-03-31T08:48:15Z Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Tutberidze, Giorgi Vilenkin system, Fejér means, martingale Hardy space, strong convergence Vilenkin system, Fejér means, martingale Hardy space, strong convergence UDC 517.5 We prove some new strong convergence theorems for partial sums and Fejér means with respect to the Vilenkin system. &nbsp; УДК 517.5 Деякi новi результати щодо строгої збiжностi середнiх Феєра вiдносно систем Вiленкiна Доведено деякі теореми про строгу збіжність часткових сум та середніх Феєра відносно системи Віленкіна. Institute of Mathematics, NAS of Ukraine 2021-04-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/226 10.37863/umzh.v73i4.226 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 4 (2021); 544 - 555 Український математичний журнал; Том 73 № 4 (2021); 544 - 555 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/226/9007 Copyright (c) 2021 Giorgi Tutberidze |
| spellingShingle | Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Persson, L.-E. Tephnadze, G. Tutberidze, G. Wall , P. Tutberidze, Giorgi Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title | Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title_alt | Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title_full | Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title_fullStr | Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title_full_unstemmed | Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title_short | Some new resuts concering strong convergence of Fejér means with respect to Vilenkin systems |
| title_sort | some new resuts concering strong convergence of fejér means with respect to vilenkin systems |
| topic_facet | Vilenkin system Fejér means martingale Hardy space strong convergence Vilenkin system Fejér means martingale Hardy space strong convergence |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/226 |
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