On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series
UDC 517.5 We establish approximation properties of Ces\`{a}ro $% (C,-\alpha ,-\beta)$ means with $\alpha ,\beta $ $\epsilon $ $(0,1)$ of Vilenkin\,--\,Fourier series. This result allows one to obtain a condition which is sufficient for the converge...
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| Дата: | 2020 |
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512248690114560 |
|---|---|
| author | Tepnadze, T. Тепнадзе, Т. Тепнадзе, Т. |
| author_facet | Tepnadze, T. Тепнадзе, Т. Тепнадзе, Т. |
| author_sort | Tepnadze, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-05-26T09:21:07Z |
| description | UDC 517.5
We establish approximation properties of Ces\`{a}ro $% (C,-\alpha ,-\beta)$ means with $\alpha ,\beta $ $\epsilon $ $(0,1)$ of Vilenkin\,--\,Fourier series. This result allows one to obtain a condition which is sufficient for the convergence of the means $\sigma _{n,m}^{-\alpha,-\beta }(x,y,f)$ to $f(x,y)$ in the $L^{p}$-metric. |
| doi_str_mv | 10.37863/umzh.v72i3.6045 |
| first_indexed | 2026-03-24T03:25:46Z |
| format | Article |
| fulltext |
UDC 517.5
T. Tepnadze (I. Javakhishvili Tbilisi State Univ., Georgia)
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS
OF NEGATIVE ORDER OF DOUBLE VILENKIN – FOURIER SERIES
ПРО АПРОКСИМАЦIЙНI ВЛАСТИВОСТI СЕРЕДНIХ ЧЕЗАРО
ВIД’ЄМНОГО ПОРЯДКУ ДЛЯ ПОДВIЙНИХ РЯДIВ ВIЛЕНКIНА – ФУР’Є
We establish approximation properties of Cesàro (C, - \alpha , - \beta ) means with \alpha , \beta \epsilon (0, 1) of Vilenkin – Fourier series. This
result allows one to obtain a condition which is sufficient for the convergence of the means \sigma - \alpha , - \beta
n,m (x, y, f) to f(x, y) in
the Lp -metric.
Для рядiв Вiленкiна – Фур’є встановлено апроксимацiйнi властивостi (C, - \alpha , - \beta ) середнiх Чезаро, \alpha , \beta \in (0, 1).
Цей результат дозволяє отримати умову, яка є достатньою для того, щоб середнi \sigma - \alpha , - \beta
n,m (x, y, f) були збiжними до
f(x, y) у метрицi Lp.
Let N+ denote the set of positive integers, N := N+ \cup \{ 0\} . Let m := (m0,m1, . . .) denote a
sequence of positive integers not lass then 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 groups Zmj ,
with the product of the discrete topologies of Zmj ’s.
The direct product of the measures
\mu k (\{ j\} ) :=
1
mk
, j \in Zmk
,
is the Haar measure on Gm with \mu (Gm) = 1. If the sequence m is bounded, then Gm is called a
bounded Vilenkin group. In this paper, we will consider only bounded Vilenkin group. The elements
of Gm can be represented by sequences x := (x0, x1, . . . , xj , . . .) , xj \in Zmj . The group operation
+ in Gm is given by
x+ y =
\bigl(
(x0 + y0)\mathrm{m}\mathrm{o}\mathrm{d}m0, . . . , (xk + yk)\mathrm{m}\mathrm{o}\mathrm{d}mk, . . .
\bigr)
,
where x := (x0, . . . , xk, . . .) and y := (y0, . . . , yk, . . .) \in Gm. The inverse of + will be denoted
by - .
It is easy to give a base for the neighborhoods of Gm :
I0(x) := Gm,
In(x) := \{ y \in Gm| y0 = x0, . . . , yn - 1 = xn - 1\}
for x \in Gm, n \in N. Define In := In (0) for n \in N+. Set en := (0, . . . , 0, 1, 0, . . .) \in Gm the
(n+ 1)th coordinate of which is 1 and the rest are zeros (n \in N) .
If we define the so-called generalized number system based on m in the following way: M0 := 1,
Mk+1 := mkMk, k \in N, then every n \in N can be uniquely expressed as n =
\sum \infty
j=0
njMj , where
nj \in Zmj , j \in N+, and only a finite number of nj ’s differ from zero. We use the following notation.
Let | n| :=max\{ k \in N : nk \not = 0\} (that is, M| n| \leq n < M| n| +1).
c\bigcirc T. TEPNADZE, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3 391
392 T. TEPNADZE
Next, we introduce of Gm an orthonormal system which is called Vilenkin system. At first define
the complex valued functions rk(x) : Gm \rightarrow C; the generalized Rademacher functions in this way
rk(x) := \mathrm{e}\mathrm{x}\mathrm{p}
2\pi ixk
mk
, i2 = - 1, x \in Gm, k \in N.
Now define the Vilenkin system \psi := (\psi n : n \in N) on Gm as follows:
\psi n(x) :=
\infty \prod
k=0
rnk
k (x), n \in N.
In particular, we call the system the Walsh – Paley if m = 2.
The Dirichlet kernels is defined by
Dn :=
n - 1\sum
k=0
\psi k, n \in N+.
Recall that (see [3] or [14])
DMn(x) =
\left\{ Mn, if x \in In,
0, if x /\in In.
(1)
The Vilenkin system is orthonormal and complete in L1 (Gm)[1].
Next, we introduce some notation with respect to the theory of two-dimensional Vilenkin system.
Let \~m be a sequence like m. The relation between the sequences ( \~mn) and
\bigl(
\~Mn
\bigr)
is the same as
between sequences (mn) and (Mn) . The group Gm \times G \~m is called a two-dimensional Vilenkin
group. The normalized Haar measure is denoted by \mu as in the one-dimensional case. We also
suppose that m = \~m and Gm \times G \~m = G2
m.
The norm of the space Lp
\bigl(
G2
m
\bigr)
is defined by
\| f\| p :=
\left( \int
G2
m
| f(x, y)| p d\mu (x, y)
\right)
1/p
, 1 \leq p <\infty .
Denote by C
\bigl(
G2
m
\bigr)
the class of continuous functions on the group G2
m, endoved with the supre-
mum norm. For the sake of brevity in notation, we agree to write L\infty \bigl( G2
m
\bigr)
instead of C
\bigl(
G2
m
\bigr)
.
The two-dimensional Fourier coefficients, the rectangular partial sums of the Fourier series, the
Dirichlet kernels with respect to the two-dimensional Vilenkin system are defined as follows:
\widehat f (n1, n2) := \int
G2
m
f(x, y) \=\psi n1(x)
\=\psi n2(y)d\mu (x, y),
Sn1,n2(x, y, f) :=
n1 - 1\sum
k1=0
n2 - 1\sum
k2=0
\widehat f(k1, k2)\psi k1(x)\psi k2(y),
Dn1,n2(x, y) := Dn1(x)Dn2(y).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 393
Denote
S(1)
n (x, y, f) :=
n - 1\sum
l=0
\widehat f(l, y)\psi l(x),
S(2)
m (x, y, f) :=
m - 1\sum
r=0
\widehat f (x, r)\psi r (y) ,
where
\widehat f(l, y) = \int
Gm
f(x, y)\psi l(x)d\mu (x)
and
\widehat f (x, r) = \int
Gm
f(x, y)\psi r (y) d\mu (y) .
The (c, - \alpha , - \beta ) means of the two-dimensional Vilenkin – Fourier series are defined as
\sigma - \alpha , - \beta
n,m (x, y, f) =
1
A - \alpha
n A - \beta
m
n\sum
i=0
m\sum
j=0
A - \alpha
n - iA
- \beta
m - j
\^f (i, j)\psi i(u)\psi j(v),
where
A\alpha
0 = 1, A\alpha
n =
(\alpha + 1) . . . (\alpha + n)
n!
.
It is well-known that [18]
A\alpha
n =
n\sum
k=0
A\alpha - 1
k , (2)
A\alpha
n - A\alpha
n - 1 = A\alpha - 1
n , (3)
A\alpha
n \sim n\alpha . (4)
The dyadic partial moduli of continuity of a function f \in Lp
\bigl(
G2
m
\bigr)
in the Lp-norm are defined
by
\omega 1
\biggl(
f,
1
Mn
\biggr)
p
= \mathrm{s}\mathrm{u}\mathrm{p}
u\in In
\| f(\cdot - u, \cdot ) - f (\cdot , \cdot )\| p ,
\omega 2
\biggl(
f,
1
Mn
\biggr)
p
= \mathrm{s}\mathrm{u}\mathrm{p}
v\in In
\| f (\cdot , \cdot - v) - f (\cdot , \cdot )\| p ,
while the dyadic mixed modulus of continuity is defined as follows:
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr)
p
=
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
394 T. TEPNADZE
= \mathrm{s}\mathrm{u}\mathrm{p}
(u,v)\in In\times Im
\| f (\cdot - u, \cdot - v) - f(\cdot - u, \cdot ) - f (\cdot , \cdot - v) + f (\cdot , \cdot )\| p ,
it is clear that
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr)
p
\leq \omega 1
\biggl(
f,
1
Mn
\biggr)
p
+ \omega 2
\biggl(
f,
1
Mm
\biggr)
p
.
The dyadic total modulus of continuity is defined by
\omega
\biggl(
f,
1
Mn
\biggr)
p
= \mathrm{s}\mathrm{u}\mathrm{p}
(u,v)\in In\times In
\| f(\cdot - u, \cdot - v) - f (\cdot , \cdot )\| p .
The problems of summability of partial sums and Cesàro means for Walsh – Fourier series were
studied in [2, 4 – 13, 16]. In his monography [17], Zhizhiashvili investigated the behavior of Cesàro
method of negative order for trigonometric Fourier series in detail. Goginava [5] studied analogical
question in case of the Walsh system. In particular, the following theorem is proved.
Theorem G [5]. Let f belongs to Lp (G2) for some p \in [1,\infty ] and \alpha \in (0, 1). Then, for any
2k \leq n < 2k+1, k, n \in N, the inequality
\bigm\| \bigm\| \sigma - \alpha
n (f) - f
\bigm\| \bigm\|
p
\leq c (p, \alpha )
\Biggl\{
2k\alpha \omega
\Bigl(
1/2k - 1, f
\Bigr)
p
+
k - 2\sum
r=0
2r - k\omega (1/2r, f)p
\Biggr\}
holds true.
In [15], the present author investigated analogous question in the case of Vilenkin system.
Theorem T. Let f belongs to Lp (Gm) for some p \in [1,\infty ] and \alpha \in (0, 1). Then, for any
Mk \leq n < Mk+1, k, n \in N, the inequality
\bigm\| \bigm\| \sigma - \alpha
n (f) - f
\bigm\| \bigm\|
p
\leq c (p, \alpha )
\Biggl\{
M\alpha
k \omega (1/Mk - 1, f)p +
k - 2\sum
r=0
Mr
Mk
\omega (1/Mr, f)p
\Biggr\}
holds true.
Goginava [7] studied approximation properties of Cesàro (c, - \alpha , - \beta ) means with \alpha , \beta \in (0, 1)
question in the case of double Walsh – Furier series. The following theorem was proved.
Theorem G2. Let f belongs to Lp
\bigl(
G2
2
\bigr)
for some p \in [1,\infty ] and \alpha , \beta \in (0, 1). Then, for any
2k \leq n < 2k+1, 2l \leq m < 2l+1, k, n \in N, the inequality\bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
n,m (f) - f
\bigm\| \bigm\| \bigm\|
p
\leq c(\alpha , \beta )
\Biggl(
2k\alpha \omega 1
\Bigl(
f, 1/2k - 1
\Bigr)
p
+ 2l\beta \omega 2
\Bigl(
f, 1/2l - 1
\Bigr)
p
+
+2k\alpha 2l\beta \omega 1,2
\Bigl(
f, 1/2k - 1, 1/2l - 1
\Bigr)
p
+
+
k - 2\sum
r=0
2r - k\omega 1 (f, 1/2
r)p +
l - 2\sum
s=0
2s - l\omega 2 (f, 1/2
s)p
\Biggr)
holds true.
In this paper, we establish analogous question in the case of double Vilenkin – Fouries series.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 395
Theorem 1. Let f belongs to Lp
\bigl(
G2
m
\bigr)
for some p \in [1,\infty ] and \alpha \in (0, 1). Then, for any
Mk \leq n < Mk+1, Ml \leq m < Ml+1, k, n,m, l \in N, the inequality
\bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
n,m (f) - f
\bigm\| \bigm\| \bigm\|
p
\leq c(\alpha , \beta )
\Biggl(
\omega 1(f, 1/Mk - 1)pM
\alpha
k + \omega 2(f, 1/Ml - 1)pM
\beta
l +
+\omega 1,2 (f, 1/Mk - 1, 1/Ml - 1)pM
\alpha
kM
\beta
l +
k - 2\sum
r=0
Mr
Mk
\omega 1 (f, 1/Mr)p +
l - 2\sum
s=0
Ms
Ml
\omega 2(f, 1/Ms)p
\Biggr)
holds true.
Corollary 1. Let f belongs to Lp for some p \in [1,\infty ]. If
M\alpha
k \omega 1
\biggl(
f,
1
Mk
\biggr)
p
\rightarrow 0 as k \rightarrow \infty , 0 < \alpha < 1,
M\beta
l \omega 1
\biggl(
f,
1
Ml
\biggr)
p
\rightarrow 0 as l \rightarrow \infty , 0 < \beta < 1,
M\alpha
kM
\beta
l \omega 1,2
\biggl(
f,
1
Mk
,
1
Ml
\biggr)
p
\rightarrow 0 as k, l \rightarrow \infty ,
then \bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
n,m (f) - f
\bigm\| \bigm\| \bigm\|
p
\rightarrow 0 as n,m\rightarrow \infty .
Corollary 2. Let f belongs to Lp for some p \in [1,\infty ] and let \alpha , \beta \in (0, 1), \alpha + \beta < 1. If
\omega
\biggl(
f,
1
Mn
\biggr)
p
= o
\Biggl( \biggl(
1
Mn
\biggr) \alpha +\beta
\Biggr)
,
then \bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
n,m (f) - f
\bigm\| \bigm\| \bigm\|
p
\rightarrow 0 as n,m\rightarrow \infty .
The following theorem shows that Corollary 2 cannot be improved.
Theorem 2. For every \alpha , \beta \in (0, 1), \alpha +\beta < 1, there exists a function f0 \in C
\bigl(
G2
m
\bigr)
for which
\omega
\biggl(
f,
1
Mn
\biggr)
C
= O
\Biggl( \biggl(
1
Mn
\biggr) \alpha +\beta
\Biggr)
,
and
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
\bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
Mn,Mn
(f) - f
\bigm\| \bigm\| \bigm\|
1
> 0.
In order to prove Theorem 1 we need the following lemmas.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
396 T. TEPNADZE
Lemma 1 [1]. Let \alpha 1, . . . , \alpha n be real numbers. Then
1
n
\int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
k=1
\alpha kDk(x)
\bigm| \bigm| \bigm| \bigm| \bigm| d\mu (x) \leq c\surd
n
\Biggl(
n\sum
k=1
\alpha 2
k
\Biggr) 1/2
,
where c is an absolute constant.
Lemma 2. Let f belongs to Lp(G2
m) for some p \in [1,\infty ]. Then, for every \alpha , \beta \in (0, 1), the
following estimation holds:
I :=
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mk - 1 - 1\sum
i=0
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq c(\alpha , \beta )
\Biggl(
k - 1\sum
r=0
Mr
Mk
\omega 1(f, 1/Mr)p +
l - 1\sum
s=0
Ms
Ml
\omega 2(f, 1/Ms)p
\Biggr)
,
where Mk \leq n < Mk+1, Ml \leq m < Ml+1.
Proof. Applying Abel’s transformation, from (2) we get
I \leq 1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mk - 1 - 1\sum
i=1
Ml - 1 - 1\sum
j=1
A - \alpha - 1
n - i+1A
- \beta - 1
m - j+1Di(u)Dj(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
A - \beta
m - Ml - 1+1DMl - 1
(v)
Mk - 1 - 1\sum
i=1
A - \alpha - 1
n - i+1Di(u)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
A - \alpha
n - Mk - 1+1DMk - 1
(u)
Ml - 1 - 1\sum
j=1
A - \beta - 1
m - j+1Dj(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 397
+
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
A - \alpha
n - Mk - 1+1A
- \beta
m - Ml - 1+1DMk - 1
(u)DMl - 1
(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
= I1 + I2 + I3 + I4. (5)
From the generalized Minkowski inequality and by (1) and (4), we obtain
I4 \leq
1
A - \alpha
n A - \beta
m
\int
G2
m
\bigm| \bigm| \bigm| A - \alpha
n - Mk - 1+1A
- \beta
m - Ml - 1+1DMk - 1
(u)DMl - 1
(v)
\bigm| \bigm| \bigm| \times
\times \| f(\cdot - u, \cdot - v) - f(x, y)\| p d\mu (u, v) \leq
\leq c(\alpha , \beta )Mk - 1Ml - 1
\int
Ik - 1\times Il - 1
\| f(\cdot - u, \cdot - v) - f(\cdot , \cdot )\| p d\mu (u, v) =
= O (\omega 1(f, 1/Mk - 1)p + \omega 2(f, 1/Ml - 1)p) . (6)
It is evident that
I1 \leq
1
A - \alpha
n A - \beta
m
k - 2\sum
r=0
l - 2\sum
s=0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mr+1 - 1\sum
i=Mr
Ms+1 - 1\sum
j=Ms
A - \alpha - 1
n - i+1A
- \beta - 1
m - j+1Di(u)Dj(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq 1
A - \alpha
n A - \beta
m
k - 2\sum
r=0
l - 2\sum
s=0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mr+1 - 1\sum
i=Mr
Ms+1 - 1\sum
j=Ms
A - \alpha - 1
n - i+1A
- \beta - 1
m - j+1Di(u)Dj(v)\times
\times
\bigl[
f(\cdot - u, \cdot - v) - SMr,Ms(\cdot - u, \cdot - v, f)
\bigr]
d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
1
A - \alpha
n A - \beta
m
k - 2\sum
r=0
l - 2\sum
s=0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mr+1 - 1\sum
i=Mr
Ms+1 - 1\sum
j=Ms
A - \alpha - 1
n - i+1A
- \beta - 1
m - j+1Di(u)Dj(v)\times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
398 T. TEPNADZE
\times
\bigl[
SMr,Ms(\cdot - u, \cdot - v, f) - SMr,Ms(\cdot , \cdot , f)
\bigr]
d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
1
A - \alpha
n A - \beta
m
k - 2\sum
r=0
l - 2\sum
s=0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mr+1 - 1\sum
i=Mr
Ms+1 - 1\sum
j=Ms
A - \alpha - 1
n - i+1A
- \beta - 1
m - j+1Di(u)Dj(v)\times
\times
\bigl[
SMr,Ms(\cdot , \cdot , f) - f(\cdot , \cdot )
\bigr]
d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
= I11 + I12 + I13. (7)
It is easy to show that
I12 = 0. (8)
By using Lemma 1, for I11, we can write
I11 \leq
1
A - \alpha
n A - \beta
m
k - 2\sum
r=0
l - 2\sum
s=0
\int
G2
m
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mr+1 - 1\sum
i=Mr
Ms+1 - 1\sum
j=Ms
A - \alpha - 1
n - i+1A
- \beta - 1
m - j+1Di(u)Dj(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \times
\times
\bigm\| \bigm\| f(\cdot - u, \cdot - v) - SMr,Ms(\cdot - u, \cdot - v, f)
\bigm\| \bigm\|
p
d\mu (u, v) \leq
\leq c(\alpha , \beta )n\alpha m\beta
k - 2\sum
r=0
l - 2\sum
s=0
(\omega 1(f, 1/Mr)p + \omega 2(f, 1/Ms)p)\times
\times
\left( \int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mr+1 - 1\sum
i=Mr
A - \alpha - 1
n - i+1Di(u)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (u)
\right) \left( \int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Ms+1 - 1\sum
j=Ms
A - \beta - 1
m - j+1Dj(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (v)
\right) \leq
\leq c(\alpha , \beta )n\alpha m\beta
k - 2\sum
r=0
l - 2\sum
s=0
(\omega 1(f, 1/Mr)p + \omega 2(f, 1/Ms)p)\times
\times
\left( \sqrt{} Mr+1
\left( Mr+1 - 1\sum
i=Mr
(n - i+ 1) - 2\alpha - 2
\right) 1/2
\right) \times
\times
\left( \sqrt{} Ms+1
\left( Ms+1 - 1\sum
j=Ms
(m - j + 1) - 2\beta - 2
\right) 1/2
\right) \leq
\leq c(\alpha , \beta )n\alpha m\beta
k - 2\sum
r=0
l - 2\sum
s=0
(\omega 1(f, 1/Mr)p + \omega 2(f, 1/Ms)p)\times
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ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 399
\times
\Bigl( \sqrt{}
Mr+1(n - Mr+1)
- \alpha - 1
\sqrt{}
Mr+1
\Bigr) \Bigl( \sqrt{}
Ms+1(n - Ms+1)
- \beta - 1
\sqrt{}
Ms+1
\Bigr)
\leq
\leq c(\alpha , \beta )n\alpha m\beta
k - 2\sum
r=0
l - 2\sum
s=0
Mr+1
M\alpha +1
k
Ms+1
M\beta +1
l
(\omega 1(f, 1/Mr)p + \omega 2(f, 1/Ms)p) \leq
\leq c(\alpha , \beta )
\Biggl(
k - 2\sum
r=0
Mr
Mk
\omega 1(f, 1/Mr)p +
l - 2\sum
s=0
Ms
Ml
\omega 2(f, 1/Ms)p
\Biggr)
. (9)
Analogously, we can prove that
I13 \leq c(\alpha , \beta )
\Biggl(
k - 2\sum
r=0
Mr
Mk
\omega 1(f, 1/Mr)p +
l - 2\sum
s=0
Ms
Ml
\omega 2(f, 1/Ms)p
\Biggr)
. (10)
Combining (7) – (10), for I1, we receive
I1 \leq c(\alpha , \beta )
\Biggl(
k - 2\sum
r=0
Mr
Mk
\omega 1(f, 1/Mr)p +
l - 2\sum
s=0
Ms
Ml
\omega 2(f, 1/Ms)p
\Biggr)
. (11)
For I2 we can write
I2 \leq
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
A - \beta
m - Ml - 1+1DMl - 1
(v)
Mk - 1 - 1\sum
i=1
A - \alpha - 1
n - i+1Di(u)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot - u, \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
A - \beta
m - Ml - 1+1DMl - 1
(v)
Mk - 1 - 1\sum
i=1
A - \alpha - 1
n - i+1Di(u)\times
\times [f(\cdot - u, \cdot ) - f(\cdot , \cdot )] d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
= I21 + I22. (12)
From the generalized Minkowski inequality and by (1) and (4), we obtain
I21 \leq c(\alpha , \beta )
Ml - 1
A - \alpha
n
\int
Il - 1
\left( \int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mk - 1 - 1\sum
i=1
A - \alpha - 1
n - i+1Di(u)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \times
\times \| f(\cdot - u, \cdot - v) - f(\cdot - u, \cdot )\| p d\mu (u)
\right) d\mu (v) \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
400 T. TEPNADZE
\leq c(\alpha , \beta )n\alpha \omega 2(f, 1/Ml - 1)
\left( \int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mk - 1 - 1\sum
i=1
A - \alpha - 1
n - i+1Di(u)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (u)
\right) \leq
\leq c(\alpha , \beta )n\alpha \omega 2(f, 1/Ml - 1)
\left( \sqrt{} Mk - 1
\left( Mk - 1 - 1\sum
i=1
(n - i+ 1) - 2\alpha - 2
\right) 1/2
\right) \leq
\leq c(\alpha , \beta )n\alpha \omega 2(f, 1/Ml - 1)
\Bigl( \sqrt{}
Mk - 1(n - Mk - 1)
- \alpha - 1
\sqrt{}
Mk - 1
\Bigr)
\leq
\leq c(\alpha , \beta )\omega 2(f, 1/Ml - 1). (13)
The estimation of I22 is analogous to the estimation of I1 and we have
I22 \leq c(\alpha , \beta )
k - 2\sum
r=0
Mr
Mk
\omega 1(f, 1/Mr)p. (14)
So, combining (12) – (14), for I2, we get
I2 \leq c(\alpha , \beta )
\Biggl(
k - 2\sum
r=0
Mr
Mk
\omega 1(f, 1/Mr)p + \omega 2(f, 1/Ml - 1)
\Biggr)
. (15)
The estimation I3 is analogous to the estimation of I2, and we obtain
I3 \leq c(\alpha , \beta )
\Biggl(
l - 2\sum
s=0
Ms
Ml
\omega 2(f, 1/Ms)p + \omega 1(f, 1/Mk - 1)
\Biggr)
. (16)
Combining (5), (6), (10), (15), (16), we receive the proof of Lemma 2.
Lemma 3. Let f belongs to Lp(G2
m) for some p \in [1,\infty ]. Then, for every \alpha , \beta \in (0, 1), the
following estimations hold:
II :=
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
n\sum
i=Mk - 1
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u)d\mu (v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq c(\alpha , \beta )\omega 1 (f, 1/Mk - 1)pM
\alpha
k ,
III :=
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
Mk - 1 - 1\sum
i=0
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u)d\mu (v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq c(\alpha , \beta )\omega 2(f, 1/Ml - 1)pM
\beta
l ,
where Mk \leq n < Mk+1, Ml \leq m < Ml+1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 401
Proof. From the generalized Minkowski inequality, we have
II =
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
n\sum
i=Mk - 1
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times f(\cdot - u, \cdot - v)d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
=
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
n\sum
i=Mk - 1
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times
\Bigl[
f(\cdot - u, \cdot - v) - S
(1)
Mk - 1
(\cdot - u, \cdot - v, f)
\Bigr]
d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq 1
A - \alpha
n A - \beta
m
\int
G2
m
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mk - 1\sum
i=Mk - 1
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \times
\times
\bigm\| \bigm\| \bigm\| f(\cdot - u, \cdot - v) - S
(1)
Mk - 1
(\cdot - u, \cdot - v, f)
\bigm\| \bigm\| \bigm\|
p
d\mu (u, v)+
+
1
A - \alpha
n A - \beta
m
\int
G2
m
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=Mk
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \times
\times
\bigm\| \bigm\| \bigm\| f(\cdot - u, \cdot - v) - S
(1)
Mk - 1
(\cdot - u, \cdot - v, f)
\bigm\| \bigm\| \bigm\|
p
d\mu (u) d\mu (v) = II1 + II2. (17)
In [15], present author showed that the inequality
\int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mk - 1\sum
v=Mk - 1
A - \alpha
n - v\psi v(u)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (u) \leq c (\alpha ) , k = 1, 2, . . . , (18)
holds true.
Using Lemma 1, by (4) and (18), for II1, we can write
II1 \leq c(\alpha , \beta )n\alpha m\beta \omega 1 (f, 1/Mk - 1)p
\left( \int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mk - 1\sum
i=Mk - 1
A - \alpha
n - i\psi i(u)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (u)
\right) \times
\times
\left( \int
Gm
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Ml - 1\sum
j=1
A - \beta
m - j+1\psi j - 1(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (v)
\right) \leq
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402 T. TEPNADZE
\leq c(\alpha , \beta )n\alpha m\beta \omega 1(f, 1/Mk - 1)p
\left( \sqrt{} Ml - 1
\left( Ml - 1\sum
i=1
(m - j + 1) - 2\beta - 2
\right) 1/2
\right) \leq
\leq c(\alpha , \beta )n\alpha m\beta \omega 2(f, 1/Mk - 1)p
\Bigl( \sqrt{}
Ml - 1 (n - Ml - 1)
- \beta - 1
\sqrt{}
Ml - 1
\Bigr)
\leq
\leq c(\alpha , \beta )\omega 1 (f, 1/Mk - 1)pM
\alpha
k . (19)
The estimation of II2 is analogous to the estimation of II1 and we have
II2 \leq c(\alpha , \beta )\omega 1(f, 1/Mk - 1)pM
\alpha
k . (20)
Combining (17) – (20), we obtain
II \leq c(\alpha , \beta )\omega 1 (f, 1/Mk - 1)pM
\alpha
k . (21)
Analogously, we can prove that
III \leq c(\alpha , \beta )\omega 2(f, 1/Ml - 1)pM
\beta
l . (22)
Combining (21), (22), we receive the proof of Lemma 3.
Lemma 4. Let f belongs to Lp(G2
m) for some p \in [1,\infty ]. Then, for every \alpha , \beta \in (0, 1), the
following estimation holds:
IV :=
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
n\sum
i=Mk - 1
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f(\cdot , \cdot )] d\mu (u)d\mu (v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq c(\alpha , \beta )\omega 1,2 (f, 1/Mk, 1/Ml)pM
\alpha
kM
\beta
l ,
where Mk \leq n < Mk+1,Ml \leq m < Ml+1.
Proof. From the generalized Minkowski inequality, and by (1) and (4), we have
IV =
1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
n\sum
i=Mk - 1
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times f(\cdot - u, \cdot - v)d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq 1
A - \alpha
n A - \beta
m
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
m
n\sum
i=Mk - 1
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times
\Bigl[
SMk - 1,Ml - 1
(\cdot - u, \cdot - v, f) - S
(1)
Mk - 1
(\cdot - u, \cdot - v, f) -
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 403
- S(2)
Ml - 1
(\cdot - u, \cdot - v, f) + f(\cdot - u, \cdot - v)
\Bigr]
d\mu (u, v)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq 1
A - \alpha
n A - \beta
m
\int
G2
m
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=Mk - 1
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \times
\times
\bigm\| \bigm\| \bigm\| SMk - 1,Ml - 1
(\cdot - u, \cdot - v, f) - S
(1)
Mk - 1
(\cdot - u, \cdot - v, f) -
- S(2)
Ml - 1
(\cdot - u, \cdot - v, f) + f(\cdot - u, \cdot - v)
\bigm\| \bigm\| \bigm\|
p
d\mu (u, v) \leq
\leq c(\alpha , \beta )n\alpha m\beta \omega 1,2 (f, 1/Mk - 1, 1/Ml - 1)p\times
\times
\int
G2
m
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
i=Mk - 1
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (u, v) \leq
\leq c(\alpha , \beta )n\alpha m\beta \omega 1,2 (f, 1/Mk - 1, 1/Ml - 1)p \leq
\leq c(\alpha , \beta )M\alpha
kM
\beta
l \omega 1,2 (f, 1/Mk - 1, 1/Ml - 1)p . (23)
Lemma 4 is proved.
Proof of Theorem 1. It is evident that
\sigma - \alpha , - \beta
n,m (f, x, y) - f(x, y) =
1
A - \alpha
n A - \beta
m
\int
G2
m
Mk - 1 - 1\sum
i=0
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f (\cdot , \cdot )] d\mu (u, v)+
+
1
A - \alpha
n A - \beta
m
\int
G2
m
n\sum
i=Mk - 1
Ml - 1 - 1\sum
j=0
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f (\cdot , \cdot )] d\mu (u, v)+
+
1
A - \alpha
n A - \beta
m
\int
G2
m
Mk - 1 - 1\sum
i=0
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f (\cdot , \cdot )] d\mu (u, v)+
+
1
A - \alpha
n A - \beta
m
\int
G2
m
n\sum
i=Mk - 1
m\sum
j=Ml - 1
A - \alpha
n - iA
- \beta
m - j\psi i(u)\psi j(v)\times
\times [f(\cdot - u, \cdot - v) - f (\cdot , \cdot )] d\mu (u)d\mu (v) = I + II + III + IV.
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404 T. TEPNADZE
Since \bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
n,m (f, x) - f(x)
\bigm\| \bigm\| \bigm\|
p
\leq \| I\| p + \| II\| p + \| III\| p + \| IV \| p .
From Lemmas 2 – 4 the proof of theorem is complete.
Proof of Corollary 2. Since
\omega i
\biggl(
f,
1
Mn
\biggr)
\leq \omega
\biggl(
f,
1
Mn
\biggr)
, i = 1, 2,
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr)
\leq 2\omega 1
\biggl(
f,
1
Mn
\biggr)
and
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr)
\leq 2\omega 2
\biggl(
f,
1
Mm
\biggr)
,
we obtain
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr)
=
\biggl(
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr) \biggr) \alpha
\alpha +\beta
\biggl(
\omega 1,2
\biggl(
f,
1
Mn
,
1
Mm
\biggr) \biggr) \beta
\alpha +\beta
\leq
\leq 2
\biggl(
\omega 1
\biggl(
f,
1
Mn
\biggr) \biggr) \alpha
\alpha +\beta
\biggl(
\omega 2
\biggl(
f,
1
Mm
\biggr) \biggr) \beta
\alpha +\beta
\leq
\leq 2
\biggl(
\omega
\biggl(
f,
1
Mn
\biggr) \biggr) \alpha
\alpha +\beta
\biggl(
\omega
\biggl(
f,
1
Mm
\biggr) \biggr) \beta
\alpha +\beta
.
The validity of Corollary 2 follows immediately from Corollary 1.
Proof of Theorem 2. First, we set
fj(x) = \rho j(x) = \mathrm{e}\mathrm{x}\mathrm{p}
2\pi ixj
mj
.
Then we define the function
f(x, y) =
\infty \sum
j=1
1
M
(\alpha +\beta )
j
fj(x)fj (y) .
First, we prove that
\omega
\biggl(
f,
1
Mn
\biggr)
C
= O
\Biggl( \biggl(
1
Mn
\biggr) \alpha +\beta
\Biggr)
. (24)
Since
| fj (x - t) - fj(x)| = 0, j = 0, 1, . . . , n - 1, t \in In,
we find
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON THE APPROXIMATION PROPERTIES OF CESÀRO MEANS OF NEGATIVE ORDER . . . 405
| f (x - t, y) - f(x, y)| \leq
n - 1\sum
j=1
1
M
(\alpha +\beta )
j
| fj (x - t) - fj(x)| +
\infty \sum
j=n
2
M
(\alpha +\beta )
j
\leq
\leq c
M
(\alpha +\beta )
n
.
Hence,
\omega 1
\biggl(
f,
1
Mn
\biggr)
= O
\Biggl( \biggl(
1
Mn
\biggr) \alpha +\beta
\Biggr)
. (25)
Analogously, we have
\omega 2
\biggl(
f,
1
Mm
\biggr)
= O
\Biggl( \biggl(
1
Mm
\biggr) \alpha +\beta
\Biggr)
. (26)
Now, by (25) and (26) , we obtain (24) .
Next, we shall prove that \sigma - \alpha , - \beta
Mn,Mn
(f) diverge in the metric of L1. It is clear that
\bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
Mn,Mn
(f) - f
\bigm\| \bigm\| \bigm\|
1
\geq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
G2
m
\Bigl[
\sigma - \alpha , - \beta
Mn,Mn
(f ;x, y) - f(x, y)
\Bigr]
\psi Mk
(x)\psi Mk
(y) d\mu (x, y)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
G2
m
\sigma - \alpha , - \beta
Mn,Mn
(f ;x, y)\psi Mk
(x)\psi Mk
(y) dxdy
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
\bigm| \bigm| \bigm| \widehat f (Mk,Mk)
\bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1
A - \alpha
Mk
A - \beta
Mk
Mlk\sum
i=0
Mlk\sum
j=0
A - \alpha
Mk - iA
- \beta
Mk - j
\^f (i, j)
\int
G2
m
\psi i(x)\psi j (y)\psi Mk
(x)\psi Mk
(y) d\mu (x, y)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
-
\bigm| \bigm| \bigm| \widehat f (Mk,Mk)
\bigm| \bigm| \bigm| = 1
A - \alpha
Mlk
A - \beta
Mlk
\bigm| \bigm| \bigm| \widehat f (Mk,Mk)
\bigm| \bigm| \bigm| - \bigm| \bigm| \bigm| \widehat f (Mk,Mk)
\bigm| \bigm| \bigm| .
We have \widehat f (Mk,Mk) =
\int
G2
m
f(x, y)\psi Mk
(x)\psi Mk
(y) d\mu (x, y) =
=
\infty \sum
j=1
1
M
(\alpha +\beta )
j
\int
G2
m
\rho j(x)\rho j (y)\psi Mk
(x)\psi Mk
(y) d\mu (x, y) =
=
\infty \sum
j=1
1
M
(\alpha +\beta )
j
\int
Gm
\rho j(x)\psi Mk
(x)d\mu (x)
\int
Gm
\rho j (y)\psi Mk
(y) d\mu (y) =
1
M
(\alpha +\beta )
k
.
So, we can write \bigm\| \bigm\| \bigm\| \sigma - \alpha , - \beta
Mn,Mn
(f) - f
\bigm\| \bigm\| \bigm\|
1
\geq c(\alpha , \beta ).
Theorem 2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
406 T. TEPNADZE
References
1. G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinshtejn, Multiplicative systems of functions and harmonic
analysis on zero-dimensional groups, Ehlm, Baku (1981) (in Russian).
2. N. J. Fine, Cesàro summability of Walsh – Fourier series, Proc. Nat. Acad. Sci. USA, 41, 558 – 591 (1995).
3. B. I. Golubov, A. V. Efimov, V. A. Skvortsov, Series and transformation of Walsh, Nauka, Moscow (1987) (in
Russian).
4. U. Goginava, On the uniform convergence of Walsh – Fourier series, Acta Math. Hungar., 93, № 1-2, 59 – 70 (2001).
5. U. Goginava, On the approximation properties of Cesàro means of negative order of Walsh – Fourier series, J.
Approxim. Theory, 115, № 1, 9 – 20 (2002).
6. U. Goginava, Uniform convergence of Cesàro means of negative order of double Walsh – Fourier series, J. Approxim.
Theory, 124, № 1, 96 – 108 (2003).
7. U. Goginava, Cesàro means of double Walsh – Fourier series, Anal. Math., 30, 289 – 304 (2004).
8. U. Goginava, K. Nagy, On the maximal operator of Walsh – Kaczmarz – Fejér means, Czechoslovak Math. J., 61(136),
№ 3, 673 – 686 (2011).
9. G. Gát, U. Goginava, A weak type inequality for the maximal operator of (C,\alpha )-means of Fourier series with respect
to the Walsh – Kaczmarz system, Acta Math. Hungar., 125, № 1-2, 65 – 83 (2009).
10. G. Gát, K. Nagy, Cesàro summability of the character system of the p-series field in the Kaczmarz rearrangement,
Anal. Math., 28, № 1, 1 – 23 (2002).
11. K. Nagy, Approximation by Cesàro means of negative order of Walsh – Kaczmarz – Fourier series, East J. Approxim.,
16, № 3, 297 – 311 (2010).
12. P. Simon, F. Weisz, Weak inequalities for Cesàro and Riesz summability of Walsh – Fourier series, J. Approxim.
Theory, 151, № 1, 1 – 19 (2008).
13. F. Schipp, Über gewisse Maximaloperatoren, Ann. Univ. Sci. Budapest. Sect. Math., 18, 189 – 195 (1975).
14. F. Schipp, W. R. Wade, P. Simon, J. Pál, Walsh series, introduction to dyadic harmonic analysis, Hilger, Bristol
(1990).
15. T. Tepnadze, On the approximation properties of Cesàro means of negative order of Vilenkin – Fourier series, Stud.
Sci. Math. Hungar., 53, № 4, 532 – 544 (2016).
16. V. I. Tevzadze, Uniform (C, - \alpha ) summability of Fourier series with respect to the Walsh – Paley system, Acta Math.
Acad. Paedagog. Nyházi. (N.S.), 22, № 1, 41 – 61 (2006).
17. L. V. Zhizhiashvili, Trigonometric Fourier series and their conjugates, Tbilisi (1993) (in Russian).
18. A. Zygmund, Trigonometric series, Vo1. 1, Cambridge Univ. Press, Cambridge, UK (1959).
Received 08.02.17,
after revision — 21.04.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
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| id | umjimathkievua-article-6045 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:46Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/56/fad110a5481e0c2ef5dfdcd838d55e56.pdf |
| spelling | umjimathkievua-article-60452020-05-26T09:21:07Z On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series Про апроксимацiйнi властивостi середнiх Чезаро вiд’ємного порядку для подвiйних рядiв Вiленкiна – Фур’є Про апроксимацiйнi властивостi середнiх Чезаро вiд’ємного порядку для подвiйних рядiв Вiленкiна – Фур’є Tepnadze, T. Тепнадзе, Т. Тепнадзе, Т. UDC 517.5 We establish approximation properties of Ces\`{a}ro $%&nbsp;(C,-\alpha ,-\beta)$ means with $\alpha ,\beta $ $\epsilon $ $(0,1)$ of&nbsp;Vilenkin\,--\,Fourier series.&nbsp;This result allows one to obtain a condition which&nbsp;is sufficient for the convergence of the means $\sigma _{n,m}^{-\alpha,-\beta }(x,y,f)$ to $f(x,y)$ in the $L^{p}$-metric. УДК 517.5 Для рядів Віленкіна\,--\,Фур'є &nbsp; встановлено апроксимаційні властивості $(C, -\alpha, -\beta)$ середніх Чезаро, $\alpha,\beta\duplicate \in (0,1).$&nbsp;Цей результат дозволяє отримати умову, яка є достатньою для того, щоб середні $\sigma_{n,m}^{-\alpha,-\beta} (x, y, f)$ були збіжними до $f(x,y)$ у метриці $L^{p}.$ Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6045 10.37863/umzh.v72i3.6045 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 3 (2020); 391-406 Український математичний журнал; Том 72 № 3 (2020); 391-406 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6045/8670 |
| spellingShingle | Tepnadze, T. Тепнадзе, Т. Тепнадзе, Т. On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series |
| title | On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series |
| title_alt | Про апроксимацiйнi властивостi середнiх Чезаро вiд’ємного порядку для подвiйних рядiв Вiленкiна – Фур’є Про апроксимацiйнi властивостi середнiх Чезаро вiд’ємного порядку для подвiйних рядiв Вiленкiна – Фур’є |
| title_full | On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series |
| title_fullStr | On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series |
| title_full_unstemmed | On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series |
| title_short | On the approximation properties of Cesàro means of negative order of double Vilenkin – Fourier series |
| title_sort | on the approximation properties of cesàro means of negative order of double vilenkin – fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6045 |
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