Strong summability of two-dimensional Vilenkin – Fourier series
We study the exponential uniform strong summability of two-dimensional Vilenkin – Fourier series. In particular, it is proved that the two-dimensional Vilenkin – Fourier series of a continuous function $f$ is uniformly strongly summable to a function $f$ exponentially in the power 1/2. Moreover, it...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507196350005248 |
|---|---|
| author | Goginava, U. Гогінава, У. |
| author_facet | Goginava, U. Гогінава, У. |
| author_sort | Goginava, U. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:55:13Z |
| description | We study the exponential uniform strong summability of two-dimensional Vilenkin – Fourier series. In particular, it is
proved that the two-dimensional Vilenkin – Fourier series of a continuous function $f$ is uniformly strongly summable to a
function $f$ exponentially in the power 1/2. Moreover, it is proved that this result is best possible. |
| first_indexed | 2026-03-24T02:05:28Z |
| format | Article |
| fulltext |
UDC 517.5
U. Goginava (Tbilisi State Univ., Georgia)
STRONG SUMMABILITY OF TWO-DIMENSIONAL
VILENKIN – FOURIER SERIES*
СИЛЬНА СУМОВНIСТЬ ДВОВИМIРНИХ РЯДIВ ВIЛЕНКIНА – ФУР’Є
We study the exponential uniform strong summability of two-dimensional Vilenkin – Fourier series. In particular, it is
proved that the two-dimensional Vilenkin – Fourier series of a continuous function f is uniformly strongly summable to a
function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.
Вивчається експоненцiальна рiвномiрна сильна сумовнiсть двовимiрних рядiв Вiленкiна – Фур’є. Зокрема, доведе-
но, що двовимiрний ряд Вiленкiна – Фур’є неперервної функцiї f є рiвномiрно сильно сумовним до функцiї f
експоненцiально в степенi 1/2. Крiм того, доведено, що цей результат є найкращим iз можливих.
1. Introduction. It is known that there exist continuous functions the trigonometric (Walsh) Fourier
series of which do not converge. However, as it was proved by Fejér’s [2] in 1905, the arithmetic
means of the differences between the function and its Fourier partial sums converge uniformly to zero.
The problem of strong summation was initiated by Hardy and Littlewood [16]. They generalized
Fejér’s result by showing that the strong means also converge uniformly to zero for any continuous
function. The investigation of the rate of convergence of the strong means was started by Alexits
[1]. Many papers have been published which are closely related with strong approximation and
summability. We note that a number of signficant results are due to Leindler [17 – 19], Totik [26 – 28],
Gogoladze [9], Goginava, Gogoladze, Karagulyan [13]. Leindler has also published the monograph
[20].
The results on strong summation and approximation of trigonometric Fourier series have been
extended for several other orthogonal systems. For instance, concerning the Walsh system see [3 – 7,
11 – 13, 21 – 24] and concerning the Ciselski system see Weisz [29, 30]. The summability of multiple
Walsh – Fourier series have been investigated in [14, 15, 31].
Fridli and Schipp [5] proved that the following is true.
Theorem FS. Let \Phi stand for the trigonometric or the Walsh system, and let \psi be a monotoni-
cally increasing function defined on [0,\infty ) for which \mathrm{l}\mathrm{i}\mathrm{m}u\rightarrow 0+ \psi (u) = 0. Then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
n\sum
k=1
\psi
\bigl( \bigm| \bigm| S\Phi
k f(x) - f(x)
\bigm| \bigm| \bigr) = 0, f \in C(G2),
if and only if there exists A > 0 such that \psi (t) \leq \mathrm{e}\mathrm{x}\mathrm{p}(At), 0 \leq t <\infty . Moreover, the convergence
is uniform in x, where S\Phi
k f denotes the kth partial sums of Fourier series of f by orthonormal
sysstem \Phi , and G2 refers to the Vilenkin group Gm with m = (2, 2, . . .).
In this paper we study the exponential uniform strong summability of two-dimensional Vilenkin –
Fourier series. In particular, it is proved that the two-dimensional Vilenkin – Fourier series of the
continuous function f is uniformly strong summable to the function f exponentially in the power
1/2. Moreover, it is proved that this result is best possible.
* The research was supported by Shota Rustaveli National Science Foundation (grant No. DI/9/5-100/13).
c\bigcirc U. GOGINAVA, 2019
340 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
STRONG SUMMABILITY OF TWO-DIMENSIONAL VILENKIN – FOURIER SERIES 341
Let \BbbN + denote the set of positive integers, \BbbN := \BbbN + \cup \{ 0\} . Let m := (m0,m1, . . .) denote a
sequence of 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 groups
Zmj , with the product of the discrete topologies of Zmj ’s. 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 the sequence m is bounded, then Gm is called a
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 = (x0 + y0(\mathrm{m}\mathrm{o}\mathrm{d}m0), . . .
. . . , xk + yk(\mathrm{m}\mathrm{o}\mathrm{d}mk), . . .), 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) :=
\bigl\{
y \in Gm | y0 = x0, . . . , yn - 1 = xn - 1
\bigr\}
for x \in Gm, n \in \BbbN . Define In := In(0) for n \in \BbbN +. Set en := (0, . . . , 0, 1, 0, . . .) \in Gm the
nth coordinate of which is 1 and the rest are zeros (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
j=0
njMj ,
where nj \in Zmj , j \in \BbbN +, and only a finite number of nj ’s differ from zero. We use the following
notation. Let (for n > 0) | n| := \mathrm{m}\mathrm{a}\mathrm{x}\{ k \in \BbbN : nk \not = 0\} (that is, M| n| \leq n < M| n| +1).
Next, we introduce on Gm an orthonormal system which is called the Vilenkin system. At first
define the complex valued functions rk(x) : Gm \rightarrow \BbbC , the generalized Rademacher functions in this
way
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 follows:
\psi n(x) :=
\infty \prod
k=0
rnk
k (x), n \in \BbbN .
Specifically, we call this system the Walsh – Paley one if m \equiv 2.
The Vilenkin system is orthonormal and complete in L1(Gm). It is well-known that \psi n(x)\psi n(y) =
= \psi n(x+ y), | \psi n(x)| = 1, n \in \BbbN , \psi n( - x) = \psi n(x) [25].
Now, introduce analogues of the usual definitions of the Fourier analysis. If f \in L1(Gm) we can
establish the following definitions in the usual way:
Fourier coefficients:
\widehat f(k) := \int
Gm
f\psi k d\mu , k \in \BbbN ,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
342 U. GOGINAVA
partial sums:
Snf :=
n - 1\sum
k=0
\widehat f(k)\psi k, n \in \BbbN +, S0f := 0,
Dirichlet kernels:
Dn :=
n - 1\sum
k=0
\psi k, n \in \BbbN +.
Recall that
DMn(x) =
\left\{ Mn, if x \in In,
0, if x \in Gm\setminus In,
(1)
Dn(x) = \psi n(x)
\infty \sum
j=0
DMj (x)
mj - 1\sum
q=mj - nj
rqj (x), f \in L1(Gm), n \in \BbbN . (2)
It is well known that
Snf(x) =
\int
Gm
f(t)Dn(x - t)d\mu (t).
Next, we introduce some notation with respect to the theory of two-dimensional Vilenkin system.
Let us fix d \geq 1, d \in \BbbN +. For Vilenkin group Gm let Gdm be its Cartesian product Gm \times . . .\times Gm
taken with itself d-times. Denote by \mu the product measure \mu \times . . .\times \mu . The rectangular partial sums
of the two-dimensional Vilenkin – Fourier series are defined as follows:
SM,N (f ;x, y) :=
M - 1\sum
i=0
N - 1\sum
j=0
\widehat f(i, j)\psi i(x)\psi j(y),
where the number \widehat f(i, j) = \int
Gm\times Gm
f(x, y)\psi i(x)\psi j(y)d\mu (x, y)
is said to be the (i, j)th Vilenkin – Fourier coefficient of the function f.
Denote
S(1)
n (f ;x, y) :=
n - 1\sum
l=0
\widehat f(l, y)\psi l(x),
S(2)
m (f ;x, y) :=
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).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
STRONG SUMMABILITY OF TWO-DIMENSIONAL VILENKIN – FOURIER SERIES 343
2. Best approximation. Denote by El,r(f) the best approximation of a function f \in C
\bigl(
G2
m
\bigr)
by Vilenkin polynomials of degree \leq l of a variable x and of degree \leq r of a variable y and let
E
(1)
l (f) be the partial best approximation of a function f \in C
\bigl(
G2
m
\bigr)
by Vilenkin polynomials of
degree \leq l of a variable x, whose coefficients are continuous functions of the remaining variable y.
Analogously, we can define E(2)
r (f).
Let ML \leq l < ML+1,MR \leq r < MR+1 and EML,MR
(f) := \| f - TML,MR
\| C , where TML,MR
is Vilenkin polynomial of best approximation of function f. Since (see (1))
\| SML,MR
(f)\| C \leq \| f\| C ,
we can write
| Sl,r(f ;x, y) - f(x, y)| \leq
\leq | Sl,r (f - SML,MR
(f);x, y)| + \| SML,MR
(f) - f\| C \leq
\leq | Sl,r (f - SML,MR
(f);x, y)| + \| SML,MR
(f - TML,MR
)\| C +
+ \| f - TML,MR
f\| C \leq
\leq | Sl,r (f - SML,MR
(f);x, y)| + 2EML,MR
(f). (3)
Now, we prove that the following inequality holds:
EML,MR
(f) \leq 2E
(1)
ML
(f) + 2E
(2)
MR
(f). (4)
Indeed, we have
EML,MR
(f) \leq \| f - SML,MR
(f)\| C =
\bigm\| \bigm\| \bigm\| f - S
(1)
ML
\Bigl(
S
(2)
MR
(f)
\Bigr) \bigm\| \bigm\| \bigm\|
C
\leq
\leq
\bigm\| \bigm\| \bigm\| f - S
(1)
ML
(f)
\bigm\| \bigm\| \bigm\|
C
+
\bigm\| \bigm\| \bigm\| S(1)
ML
\Bigl(
S
(2)
MR
(f) - f
\Bigr) \bigm\| \bigm\| \bigm\|
C
\leq
\leq
\bigm\| \bigm\| \bigm\| f - S
(1)
ML
(f)
\bigm\| \bigm\| \bigm\|
C
+
\bigm\| \bigm\| \bigm\| S(2)
MR
(f) - f
\bigm\| \bigm\| \bigm\|
C
. (5)
Let T (1)
ML
(x, y) be a polynomial of the best approximation E(1)
ML
(f). Then\bigm\| \bigm\| \bigm\| S(1)
ML
(f) - f
\bigm\| \bigm\| \bigm\|
C
\leq
\bigm\| \bigm\| \bigm\| f - T
(1)
ML
\bigm\| \bigm\| \bigm\|
C
+
\bigm\| \bigm\| \bigm\| S(1)
ML
\Bigl(
f - T
(1)
ML
\Bigr) \bigm\| \bigm\| \bigm\|
C
\leq
\leq 2
\bigm\| \bigm\| \bigm\| f - T
(1)
ML
\bigm\| \bigm\| \bigm\|
C
= 2E
(1)
ML
(f). (6)
Analogously, we can prove that \bigm\| \bigm\| \bigm\| S(2)
MR
(f) - f
\bigm\| \bigm\| \bigm\|
C
\leq 2E
(2)
MR
(f). (7)
Combining (5) – (7) we obtain (4).
It is easy to show that
\| f - SMLMR
(f)\| C \leq 2EML,MR
(f). (8)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
344 U. GOGINAVA
3. Main results.
Theorem 1. Let f \in C
\bigl(
G2
m
\bigr)
. Then the inequality\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 1
nm
n\sum
l=1
m\sum
r=1
\biggl(
eA| Sl,r(f) - f| 1/2 - 1
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
C
\leq
\leq c(f,A)
n
n\sum
l=1
\sqrt{}
E
(1)
l (f) +
c(f,A)
m
m\sum
r=1
\sqrt{}
E
(2)
r (f)
is satisfied for any A > 0, where c(f,A) is a positive constant depend on A and f.
We say that the function \psi belongs to the class \Psi if it increases on [0,+\infty ) and
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow 0
\psi (u) = \psi (0) = 0.
Theorem 2. (a) Let \varphi \in \Psi and the inequality
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
\varphi (u)\surd
u
<\infty (9)
holds. Then for any function f \in C
\bigl(
G2
m
\bigr)
the equality
\mathrm{l}\mathrm{i}\mathrm{m}
n,m\rightarrow \infty
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 1
nm
n\sum
l=1
m\sum
r=1
\Bigl(
e\varphi (| Sl,r(f) - f| ) - 1
\Bigr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
C
= 0
is satisfied.
(b) For any function \varphi \in \Psi satisfying the condition
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
\varphi (u)\surd
u
= \infty (10)
there exists a function F \in C
\bigl(
G2
m
\bigr)
such that
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
1
m2
m\sum
l=1
m\sum
r=1
e\varphi (| Sl,r(F ;0,0) - F (0,0)| ) = +\infty .
4. Auxiliary results.
Lemma 1 [8]. Let p \in \BbbN +. Then
\mathrm{s}\mathrm{u}\mathrm{p}
n
\left( \int
Gp
m
1
Mn
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Mn+1 - 1\sum
l=Mn
p\prod
k=1
Dl (sk)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (s1, . . . , sp)
\right)
1/p
\leq cp,
where c is a positive constant.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
STRONG SUMMABILITY OF TWO-DIMENSIONAL VILENKIN – FOURIER SERIES 345
Lemma 2 [9]. Let \varphi ,\psi \in \Psi and the equality
\mathrm{l}\mathrm{i}\mathrm{m}
n,m\rightarrow \infty
1
nm
n\sum
l=1
m\sum
r=1
\psi
\bigl(
| Sl,r(f ;x, y) - f(x, y)|
\bigr)
= 0
be satisfied at the point (x0, y0) or uniformly on a set E \subset I2. If
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
\varphi (u)
\psi (u)
<\infty ,
then the equality
\mathrm{l}\mathrm{i}\mathrm{m}
n,m\rightarrow \infty
1
nm
n\sum
l=1
m\sum
r=1
\varphi (| Sl,r(f ;x, y) - f(x, y)| ) = 0
is satisfied at the point (x0, y0) or uniformly on a set E \subset I2.
Lemma 3. Let p > 0, A, B \in \BbbN . Then\left\{ 1
MAMB
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
| Sn,l(f ;x, y)| p
\right\}
1/p
\leq c \| f\| C (p+ 1)2. (11)
Proof. Since \left\{ 1
MAMB
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
| Sn,l(f ;x, y)| p
\right\}
1/p
\leq
\leq
\left\{ 1
MAMB
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
| Sn,l(f ;x, y)| p+1
\right\}
1/(p+1)
without lost of generality we can suppose that p = 2m, m \in \BbbN +. We can write
| Sn,l(f ;x, y)| 2 = Sn,l(f ;x, y)Sn,l(f ;x, y) =
=
\int
G2
m
f(x - s1, y - t1)Dn(s1)Dl(t1)d\mu (s1, t1)\times
\times
\int
G2
m
f (x - s2, y - t2)Dn(s2)Dl(t2)d\mu (s2, t2) =
=
\int
G2
m
f(x - s1, y - t1)Dn(s1)Dl(t1)d\mu (s1, t1)\times
\times
\int
G2
m
f(x+ s2, y + t2)Dn(s2)Dl(t2)d\mu (s2, t2) =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
346 U. GOGINAVA
=
\int
G4
m
f(x - s1, y - t1)f (x+ s2, y + t2)\times
\times Dn(s1)Dn(s2)Dl(t1)Dl(t2)d\mu (s1, t1, s2, t2).
Hence, we get
| Sn,l(f ;x, y)| p =
\Bigl(
| Sn,l(f ;x, y)| 2
\Bigr) p/2
=
=
\left( \int
G4
m
f(x - s1, y - t1)f(x+ s2, y + t2)\times
\times Dn(s1)Dn(s2)Dl(t1)Dl(t2)d\mu (s1, t1, s2, t2)
\right)
p/2
=
=
\int
G2p
m
p/2\prod
k=1
f (x - s2k - 1, y - t2k - 1)
p/2\prod
r=1
f(x+ s2r, y + t2r)\times
\times
p\prod
i=1
Dn(si)
p\prod
j=1
Dl(tj)d\mu (s1, t1, . . . , sp, tp) ,
\left\{ 1
MAMB
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
| Sn,l(f ;x, y)| p
\right\}
1/p
\leq
\leq
\left( \int
G2p
m
p/2\prod
k=1
| f (x - s2k - 1, y - t2k - 1)|
p/2\prod
r=1
\bigm| \bigm| f(x+ s2r, y + t2r)
\bigm| \bigm| \times
\times 1
MAMB
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
p\prod
i=1
Dn(si)
p\prod
j=1
Dl(tj)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (s1, t1, . . . , sp, tp)
\right) 1/p
\leq
\leq \| f\| C
\left( \int
Gp
m
1
MA
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
MA+1 - 1\sum
n=MA
p\prod
i=1
Dn(si)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (s1, . . . , sp)
\right)
1/p
\times
\times
\left( \int
Gp
m
1
MB
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
MB+1 - 1\sum
l=MB
p\prod
j=1
Dl(tj)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (t1, . . . , tp)
\right)
1/p
\leq
\leq cp2 \| f\| C .
Lemma 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
STRONG SUMMABILITY OF TWO-DIMENSIONAL VILENKIN – FOURIER SERIES 347
Lemma 4. Let f \in C
\bigl(
G2
m
\bigr)
and p > 0. Then
1
nk
n\sum
l=1
k\sum
r=1
\bigm| \bigm| Sl,r(f ;x, y) - f(x, y)
\bigm| \bigm| p \leq
\leq cp(p+ 1)2p
\Biggl\{
1
n
n\sum
l=1
\Bigl(
E
(1)
l (f)
\Bigr) p
+
1
k
k\sum
r=1
\Bigl(
E(2)
r (f)
\Bigr) p\Biggr\}
. (12)
Proof. Since
(a+ b)\beta \leq 2\beta
\bigl(
a\beta + b\beta
\bigr)
, \beta > 0,
from (3), (4), (8) and using Lemma 3 we get
1
MAMB
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
| Sn,l(f ;x, y) - f(x, y)| p \leq
\leq 2p
MAMB
MA+1 - 1\sum
n=MA
MB+1 - 1\sum
l=MB
| Sn,l (f - SMAMB
(f);x, y)| p+
+
22p
MAMB
(MA+1 - MA) (MB+1 - MB)E
p
MAMB
(f) \leq
\leq cp(p+ 1)2p \| f - SMAMB
(f)\| pC +
+cp
\Bigl( \Bigl(
E
(1)
MA
(f)
\Bigr) p
+
\Bigl(
E
(2)
MB
(f)
\Bigr) p\Bigr)
\leq
\leq cp(p+ 1)2p
\Bigl( \Bigl(
E
(1)
MA
(f)
\Bigr) p
+
\Bigl(
E
(2)
MB
(f)
\Bigr) p\Bigr)
. (13)
Let ML \leq n < ML+1 and MR \leq k < MR+1. Then from (13) we have
1
nk
n\sum
l=1
k\sum
r=1
\bigm| \bigm| Sl,r(f ;x, y) - f(x, y)
\bigm| \bigm| p \leq
\leq 1
nk
ML+1 - 1\sum
l=1
MR+1 - 1\sum
r=1
| Sl,r(f ;x, y) - f(x, y)| p =
=
1
nk
L\sum
A=0
R\sum
B=0
MA+1 - 1\sum
l=MA
MB+1 - 1\sum
r=MB
\bigm| \bigm| Sl,r(f ;x, y) - f(x, y)
\bigm| \bigm| p \leq
\leq cp(p+ 1)2p
nk
MAMB
L\sum
A=0
R\sum
B=0
\Bigl( \Bigl(
E
(1)
MA
(f)
\Bigr) p
+
\Bigl(
E
(2)
Mb
(f)
\Bigr) p\Bigr)
\leq
\leq cp(p+ 1)2p
nk
L\sum
A=0
R\sum
B=0
MA - 1\sum
l=MA - 1
MB - 1\sum
r=MB - 1
\Bigl( \Bigl(
E
(1)
MA
(f)
\Bigr) p
+
\Bigl(
E
(2)
MB
(f)
\Bigr) p\Bigr)
\leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
348 U. GOGINAVA
\leq cp(p+ 1)2p
nk
L\sum
A=0
R\sum
B=0
MA - 1\sum
l=MA - 1
MB - 1\sum
r=MB - 1
\Bigl( \Bigl(
E
(1)
l (f)
\Bigr) p
+
\Bigl(
E(2)
r (f)
\Bigr) p\Bigr)
\leq
\leq cp(p+ 1)2p
nk
n\sum
l=1
k\sum
r=1
\Bigl( \Bigl(
E
(1)
l (f)
\Bigr) p
+
\Bigl(
E(2)
r (f)
\Bigr) p\Bigr)
\leq
\leq cp(p+ 1)2p
\Biggl\{
1
n
n\sum
l=1
\Bigl(
E
(1)
l (f)
\Bigr) p
+
1
k
k\sum
r=1
\Bigl(
E(2)
r (f)
\Bigr) p\Biggr\}
.
Lemma 4 is proved.
5. Proofs of main results. The Walsh – Paley version of Theorem 1 were proved in [12]. Based
on inequality (12) the same construction works for the Vilenkin case. Therefore the proof of Theorem
1 will be omitted.
Proof of Theorem 2. (a) It is easy to see that if \varphi \in \Psi , then e\varphi - 1 \in \Psi . Besides, (9) implies
the existence of a number A such that
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
e\varphi (u) - 1
eAu
1/2 - 1
<\infty .
Therefore, in view of Lemma 2, for the proof of Theorem 2 it is sufficient to prove that
\mathrm{l}\mathrm{i}\mathrm{m}
n,m\rightarrow \infty
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 1
nm
n\sum
l=1
m\sum
r=1
\biggl(
eA| Sl,r(f) - f| 1/2 - 1
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
C
= 0. (14)
The validity of equality of (14) immediately follows from Theorem 1.
(b) First of all we prove that if \psi \in \Psi and
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
\psi (u)
u
= \infty
then there exists a function f \in C(Gm) and sequence of positive integers \{ Ak : k \geq 1\} such that
\psi
\Bigl( \bigm| \bigm| \bigm| SNAk
(f, 0)
\bigm| \bigm| \bigm| \Bigr) > 5(Ak - 1) \mathrm{l}\mathrm{n} a, (15)
where a := \mathrm{s}\mathrm{u}\mathrm{p}
j
mj and NAj :=
\sum Aj - 1
k=Aj - 1
\Bigl[ m2k
2
\Bigr]
M2k.
Let \{ Bk : k \geq 1\} be an increasing sequence of positive integers such that
B1 > c\prime , (16)
Bj > 2Bj - 1, (17)
\psi (Bj)
Bj
>
5j \mathrm{l}\mathrm{n} a
c\prime
, (18)
where the constant c\prime would be definen below.
Set
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
STRONG SUMMABILITY OF TWO-DIMENSIONAL VILENKIN – FOURIER SERIES 349
Ak :=
\biggl[
kBk
c\prime
\biggr]
+ 1,
fj(x) :=
1
j + 1
Aj - 1\sum
s=Aj - 1
m2s+1 - 1\sum
x2s+1=0
. . .
m2Aj
- 1\sum
x2Aj - 1=0
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl(
- i \mathrm{a}\mathrm{r}\mathrm{g}
\Bigl(
DNAj
(x)
\Bigr) \Bigr)
\times
\times \BbbI
I2Aj
\Bigl(
0,...,0,x2s=m2s - 1,x2s+1,...,x2Aj - 1
\Bigr) (x),
f(x) :=
\infty \sum
j=1
fj(x), f (0) = 0,
where \BbbI E is characteristic function of the set E \subset Gm.
Since
I2Aj
\bigl(
0, . . . , 0, x2s = m2s - 1, x2s+1, . . . , x2Aj - 1
\bigr)
\cap
\cap I2Aj
\bigl(
0, . . . , 0, x2l = m2l - 1, x2l+1, . . . , x2Aj - 1
\bigr)
= \varnothing , l \not = s,
and 1/(j + 1) \rightarrow 0 as j \rightarrow \infty we conclude that f \in C(Gm).
We can write \bigm| \bigm| \bigm| SNAk
(f ; 0) - f(0)
\bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
Gm
f(t)DNAk
(t)d\mu (t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \geq
\geq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
Gm
fk(t)DNAk
(t)d\mu (t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
\infty \sum
j=k+1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
Gm
fj (t)DNAk
(t)d\mu (t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| -
-
k - 1\sum
j=0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
Gm
fj(t)DNAk
(t)d\mu (t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
= J1 - J2 - J3. (19)
From the definition of the function f we have
J1 =
1
k + 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
Ak - 1\sum
s=Ak - 1
m2s+1 - 1\sum
t2s+1=0
. . .
. . .
m2Ak
- 1\sum
t2Ak - 1=0
\int
I2Ak(0,...,0,t2s=m2s - 1,t2s+1,...,t2Ak - 1)
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl(
- i \mathrm{a}\mathrm{r}\mathrm{g}
\Bigl(
DNAk
(t)
\Bigr) \Bigr)
DNAk
(t)d\mu (t)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
350 U. GOGINAVA
=
1
k + 1
Ak - 1\sum
s=Ak - 1
m2s+1 - 1\sum
t2s+1=0
. . .
. . .
m2Ak
- 1\sum
t2Ak - 1=0
\int
I2Ak(0,...,0,t2s=m2s - 1,t2s+1,...,t2Ak - 1)
\bigm| \bigm| \bigm| DNAk
(t)
\bigm| \bigm| \bigm| d\mu (t).
Since (see [10]) \bigm| \bigm| \bigm| DNAk
(t)
\bigm| \bigm| \bigm| \geq cM2s+1
for
t \in I2s+1 (0, . . . , 0, t2s = m2s - 1) , s = Ak - 1, . . . , Ak - 1,
from (17) we can write
J1 \geq
c
k + 1
Ak - 1\sum
s=Ak - 1
M2s+1
m2s+1 - 1\sum
t2s+1=0
. . .
. . .
m2Ak - 1 - 1\sum
t2Ak - 1=0
\mu (I2Ak
(0, . . . , 0, t2s = m2s - 1, t2s+1, . . . , t2Ak - 1))=
=
c
k + 1
Ak - 1\sum
s=Ak - 1
M2s+1m2s+1 . . .m2Ak - 1
M2Ak
=
=
c
k + 1
(Ak - Ak - 1) .
Since (see (16))
Ak - Ak - 1 =
\biggl[
kBk
c\prime
\biggr]
-
\biggl[
(k - 1)Bk - 1
c\prime
\biggr]
\geq
\geq kBk
c\prime
- (k - 1)Bk - 1
c\prime
- 1 =
=
k (Bk - Bk - 1)
c\prime
+
Bk - 1
c\prime
- 1 >
>
k (Bk - Bk - 1)
c\prime
>
kBk
2c\prime
\geq Ak
2
- 1
2
>
Ak
4
,
for J1 we have
J1 \geq
c
k + 1
Ak
4
.
For J2 we get
J2 \leq
\infty \sum
j=k+1
1
j + 1
Aj - 1\sum
s=Aj - 1
1
M2s
NAk
\leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
STRONG SUMMABILITY OF TWO-DIMENSIONAL VILENKIN – FOURIER SERIES 351
\leq 1
k
\infty \sum
s=Ak
1
M2s
NAk
\leq c
k
. (20)
By (2) and from the construction of the function fj we can write
supp (fj) \cap supp
\Bigl(
DNAk
\Bigr)
= \varnothing , j = 1, 2, . . . , k - 1,
consequently,
J3 = 0. (21)
Combining (18) – (21) we conclude that\bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| SNAk
(f ; 0) - f(0)
\bigm| \bigm| \bigm| \geq c\prime Ak
k
\geq Bk,
\psi
\Bigl( \bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| \Bigr) \geq \psi (Bk) \geq
5k \mathrm{l}\mathrm{n} a
c\prime
Bk \geq 5(Ak - 1) \mathrm{l}\mathrm{n} a.
Hence, (15) is proved.
Write \varphi (u) = \lambda (u)
\surd
u and define \psi (u) := \lambda
\bigl(
u2
\bigr)
u. Then
\mathrm{l}\mathrm{i}\mathrm{m}
u\rightarrow \infty
\psi (u)
u
= +\infty .
Therefore there exist a function f \in C(Gm) and sequence of positive integers \{ Ak : k \geq 1\} for
which
\psi
\Bigl( \bigm| \bigm| \bigm| SNAk
(f, 0)
\bigm| \bigm| \bigm| \Bigr) > 5(Ak - 1) \mathrm{l}\mathrm{n} a. (22)
Set
F (x, y) := f(x)f(y).
It is easy to show that
\varphi
\Bigl( \bigm| \bigm| \bigm| SNAk
,NAk
(F ; 0, 0)
\bigm| \bigm| \bigm| \Bigr) = \varphi
\biggl( \bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| 2\biggr) =
= \lambda
\biggl( \bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| 2\biggr) \bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| = \psi
\Bigl( \bigm| \bigm| \bigm| SNAk
(f ; 0)
\bigm| \bigm| \bigm| \Bigr) .
Since NAk
\leq a2Ak , from (22) we have
1
N2
Ak
NAk\sum
i=1
NAk\sum
j=1
e\varphi (| Si,j(F ;0,0)| ) \geq 1
N2
Ak
e
\varphi
\Bigl( \bigm| \bigm| \bigm| SNAk
,NAk
(F ;0,0)
\bigm| \bigm| \bigm| \Bigr)
=
=
1
N2
Ak
e
\psi
\Bigl( \bigm| \bigm| \bigm| SNAk
(f ;0)
\bigm| \bigm| \bigm| \Bigr) \geq e5(Ak - 1) ln a
a4Ak
\rightarrow \infty as k \rightarrow \infty .
Theorem 2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
352 U. GOGINAVA
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ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 3
|
| id | umjimathkievua-article-1443 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:28Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/35/9f61ac6e25b5cb13d20617fc397e1c35.pdf |
| spelling | umjimathkievua-article-14432019-12-05T08:55:13Z Strong summability of two-dimensional Vilenkin – Fourier series Сильна сумовнiсть двовимiрних рядiв Вiленкiна – Фур’є Goginava, U. Гогінава, У. We study the exponential uniform strong summability of two-dimensional Vilenkin – Fourier series. In particular, it is proved that the two-dimensional Vilenkin – Fourier series of a continuous function $f$ is uniformly strongly summable to a function $f$ exponentially in the power 1/2. Moreover, it is proved that this result is best possible. Вивчається експоненцiальна рiвномiрна сильна сумовнiсть двовимiрних рядiв Вiленкiна – Фур’є. Зокрема, доведено, що двовимiрний ряд Вiленкiна – Фур’є неперервної функцiї $f$ є рiвномiрно сильно сумовним до функцiї $f$ експоненцiально в степенi 1/2. Крiм того, доведено, що цей результат є найкращим iз можливих Institute of Mathematics, NAS of Ukraine 2019-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1443 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 3 (2019); 340-352 Український математичний журнал; Том 71 № 3 (2019); 340-352 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1443/427 Copyright (c) 2019 Goginava U. |
| spellingShingle | Goginava, U. Гогінава, У. Strong summability of two-dimensional Vilenkin – Fourier series |
| title | Strong summability of two-dimensional Vilenkin – Fourier series |
| title_alt | Сильна сумовнiсть двовимiрних рядiв Вiленкiна – Фур’є |
| title_full | Strong summability of two-dimensional Vilenkin – Fourier series |
| title_fullStr | Strong summability of two-dimensional Vilenkin – Fourier series |
| title_full_unstemmed | Strong summability of two-dimensional Vilenkin – Fourier series |
| title_short | Strong summability of two-dimensional Vilenkin – Fourier series |
| title_sort | strong summability of two-dimensional vilenkin – fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1443 |
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