On the summability of double Walsh - Fourier series of functions of bounded generalized variation
The convergence of Cesaro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated.
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| author | Goginava, U. Гогінава, У. |
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| description | The convergence of Cesaro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated. |
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UDC 517.5
U. Goginava (Tbilisi State Univ., Georgia)
ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES
OF FUNCTIONS OF BOUNDED GENERALIZED VARIATION
ПРО СУМОВНIСТЬ ПОДВIЙНИХ РЯДIВ УОЛША – ФУР’Є
ФУНКЦIЙ ОБМЕЖЕНОЇ УЗАГАЛЬНЕНОЇ ВАРIАЦIЇ
The convergence of Cesàro means of negative order of double Walsh – Fourier series of functions of bounded generalized
variation is investigated.
Дослiджується збiжнiсть середнiх Чезаро вiд’ємного порядку вiд подвiйних рядiв Уолша – Фур’є функцiй обмеженої
узагальненої варiацiї.
1. Classes of functions of bounded generalized variation. In 1881 Jordan [14] introduced a class
of functions of bounded variation and applied it to the theory of Fourier series. Hereinafter this notion
was generalized by many authors (quadratic variation, Φ-variation, Λ-variation etc., see [2, 15, 25,
23]). In two dimensional case the class BV of functions of bounded variation was introduced by
Hardy [13].
Let f be a real and measurable function of two variable of period 2π with respect to each variable.
Given intervals ∆ = (a, b), J = (c, d) and points x, y from I : = [0, 1) we denote
f(∆, y) : = f(b, y)− f(a, y), f(x, J) = f(x, d)− f(x, c)
and
f(∆, J) : = f(a, c)− f(a, d)− f(b, c) + f(b, d).
Let E = {∆i} be a collection of nonoverlapping intervals from I ordered in arbitrary way and let Ω
be the set of all such collections E.
For the sequence of positive numbers Λ1 = {λ1
n}∞n=1, Λ2 = {λ2
n}∞n=1 and I2 : = [0, 1)2 we
denote
Λ1V1(f ; I2) = sup
y
sup
E∈Ω
∑
i
|f(∆i, y)|
λ1
i
(E = {∆i}),
Λ2V2(f ; I2) = sup
x
sup
F∈Ω
∑
j
|f(x, Jj)|
λ2
j
(F = {Jj}),
(
Λ1Λ2
)
V1,2(f ; I2) = sup
F,E∈Ω
∑
i
∑
j
|f(∆i, Jj)|
λ1
iλ
2
j
.
Definition 1. We say that the function f has bounded
(
Λ1,Λ2
)
-variation on I2 and write
f ∈
(
Λ1,Λ2
)
BV
(
I2
)
, if(
Λ1,Λ2
)
V (f ; I2) := Λ1V1(f ; I2) + Λ2V2(f ; I2) +
(
Λ1Λ2
)
V1,2(f ; I2) <∞.
We say that the function f has bounded partial Λ-variation and write f ∈ PΛBV
(
I2
)
if
PΛBV (f ; I2) := ΛV1(f ; I2) + ΛV2(f ; I2) <∞.
c© U. GOGINAVA, 2012
490 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 491
If λn ≡ 1 (or if 0 < c < λn < C <∞, n = 1, 2, . . . ) the class PΛBV coincide with the PBV
class of bounded partial variation introduced by Goginava [7]. Hence it is reasonable to assume that
λn → ∞ and since the intervals in E = {∆i} are ordered arbitrarily, we will suppose, without loss
of generality, that the sequence {λn} is increasing. Thus,
1 < λ1 ≤ λ2 ≤ . . . , lim
n→∞
λn =∞. (1)
We also suppose that
∑∞
n=1
(1/λn) = +∞.
In the case when λn = n, n = 1, 2, . . . , we say harmonic variation instead of Λ-variation and
write H instead of Λ (HBV , PHBV , HV (f), etc.).
The notion of Λ-variation was introduced by Waterman [23] in one dimensional case, by Sahakian
[20] in two dimensional case. The notion of bounded partial Λ-variation (PΛBV ) was introduced by
Goginava and Sahakian [11].
Definition 2. We say that the function f is continuous in
(
Λ1,Λ2
)
-variation on I2 and write
f ∈ C
(
Λ1,Λ2
)
V
(
I2
)
, if
lim
n→∞
Λ1
nV1
(
f ; I2
)
= lim
n→∞
Λ2
nV2
(
f ; I2
)
= 0
and
lim
n→∞
(
Λ1
n,Λ
2
)
V1,2
(
f ; I2
)
= lim
n→∞
(
Λ1,Λ2
n
)
V1,2
(
f ; I2
)
= 0,
where Λin :=
{
λik
}∞
k=n
=
{
λik+n
}∞
k=0
, i = 1, 2.
2. Walsh function. Let P denote the set of positive integers, N : = P ∪ {0}. The set of all
integers by Z and the set of dyadic rational numbers in the unit interval I : = [0, 1) by Q. In
particular, each element of Q has the form
p
2n
for some p, n ∈ N, 0 ≤ p ≤ 2n. By a dyadic interval
in I we mean one of the form I lN : = [l2−N , (l + 1) 2−N ) for some l ∈ N, 0 ≤ l < 2N . Given
N ∈ N and x ∈ I , let IN (x) denote a dyadic interval of length 2−N which contains the point x.
Denote IN : = [0, 2−N ) and IN : = I\IN . Set (i, j) ≤ (n,m) if i ≤ n and j ≤ m.
Let r0 (x) be the function defined by
r0 (x) =
{
1, if x ∈ [0, 1/2),
−1, if x ∈ [1/2, 1),
r0 (x+ 1) = r0 (x) .
The Rademacher system is defined by
rn (x) = r0 (2nx) , n ≥ 1.
Let w0, w1, . . . represent the Walsh functions, i.e., w0 (x) = 1 and if k = 2n1 + . . . + 2ns is a
positive integer with n1 > n2 > . . . > ns then
wk (x) = rn1 (x) . . . rns (x) .
The Walsh – Dirichlet kernel is defined by
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
492 U. GOGINAVA
Dn (x) =
n−1∑
k=0
wk (x) .
Recall that [12, 21]
D2n (x) =
{
2n, if x ∈ [0, 2−n) ,
0, if x ∈ [2−n, 1) ,
(2)
and
D2n+m (x) = D2n (x) + w2n (x)Dm (x) , 0 ≤ m < 2n. (3)
It is well known that [21]
Dn (t) = wn (t)
∞∑
j=0
njw2j (t)D2j (t) , (4)
where n =
∑∞
j=0
nj2
j . Denote for n ∈ P, |n| : = max{j ∈ N : nj 6= 0}, that is 2|n| ≤ n < 2|n|+1.
Given x ∈ I , the expansion
x =
∞∑
k=0
xk2
−(k+1), (5)
where each xk = 0 or 1, will be called a dyadic expansion of x. If x ∈ I\Q , then (5) is uniquely
determined. For the dyadic expansion x ∈ Q we choose the one for which limk→∞ xk = 0.
The dyadic sum of x, y ∈ I in terms of the dyadic expansion of x and y is defined by
xu y =
∞∑
k=0
|xk − yk| 2−(k+1).
We say that f (x, y) is continuous at (x, y) if
lim
h,δ→0
f (xu h, y u δ) = f (x, y) . (6)
Set
ω (f ; IM (x)× IN (y)) : = sup
(s,t)∈IM×IN
|f (xu s, y u t)− f (x, y)| .
We consider the double system {wn(x)× wm(y) : n,m ∈ N} on the unit square I2 = [0, 1) ×
× [0, 1) .
If f ∈ L1
(
I2
)
, then
f̂ (n,m) =
∫
I2
f (x, y)wn(x)wm(y)dxdy
is the (n,m)-th Walsh – Fourier coefficient of f.
The rectangular partial sums of double Fourier series with respect to the Walsh system are defined
by
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 493
SM,Nf(x, y) =
M−1∑
m=0
N−1∑
n=0
f̂ (m,n)wm(x)wn(y).
The Cesàro (C;α, β)-means of double Walsh – Fourier series are defined as follows:
σα,βn,mf (x, y) =
1
Aαn−1A
β
m−1
n∑
i=1
m∑
j=1
Aα−1
n−iA
β−1
m−jSi,jf (x, y) ,
where
Aα0 = 1, Aαn =
(α+ 1) . . . (α+ n)
n!
, α 6= −1,−2, . . . .
It is well-known that [27]
Aαn =
n∑
k=0
Aα−1
n−k, (7)
Aαn ∼ nα (8)
and
σα,βn,mf (x, y) =
∫
I2
f (s, t)Kα
n (xu s)Kβ
m (y u t) dsdt,
where
Kα
n (x) :=
1
Aαn−1
n∑
k=1
Aα−1
n−kDk (x) .
Given a function f (x, y) , periodic in both variables with period 1, for 0 ≤ j < 2m and 0 ≤ i <
< 2n and integers m, n ≥ 0 we set
∆m
j f (x, y)1 = f
(
xu 2j2−m−1, y
)
− f
(
xu (2j + 1) 2−m−1, y
)
,
∆n
i f (x, y)2 = f
(
x, y u 2i2−n−1
)
− f
(
x, y u (2i+ 1) 2−n−1
)
,
∆mn
ji f (x, y) = ∆n
i
(
∆m
j f (x, y)1
)
2
= ∆m
j (∆n
i f (x, y)2)1 =
= f
(
xu 2j2−m−1, y u 2i2−n−1
)
− f
(
xu (2j + 1) 2−m−1, y u 2i2−n−1
)
−
−f
(
xu 2j2−m−1, y u (2i+ 1) 2−n−1
)
+
+f
(
xu (2j + 1) 2−m−1, y u (2i+ 1) 2−n−1
)
.
3. Formulation of problems. The well known Dirichlet – Jordan theorem (see [27]) states that
the Fourier series of a function f(x), x ∈ T of bounded variation converges at every point x to the
value [f (x+ 0) + f (x− 0)] /2.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
494 U. GOGINAVA
Hardy [13] generalized the Dirichlet – Jordan theorem to the double Fourier series. He proved
that if function f(x, y) has bounded variation in the sense of Hardy (f ∈ BV ), then S [f ] converges
at any point (x, y) to the value
1
4
∑
f (x± 0, y ± 0).
Convergence of rectangular and spherical partial sums of d-dimensional trigonometric Fourier
series of functions of bounded Λ-variation was investigated in details by Sahakian [20], Dyachenko
[4, 5, 6], Bakhvalov [1], Sablin [19].
For the two-dimensional Walsh – Fourier series the convergence of partial sums of functions
Harmonic bounded fluctuation and other bounded generalized variation were studied by Moricz
[16, 17], Onnewer, Waterman [18], Waterman [24], Goginava [8, 9].
For the two-dimensional Walsh – Fourier series the summability by Cesáro method of negative
order for functions of partial bounded variation investigated by the author.
Theorem G1 (Goginava [10]). Let f ∈ Cw
(
I2
)
∩ PBV and α + β < 1, α, β > 0. Then the
double Walsh – Fourier series of the function f is uniformly (C;−α,−β) summable in the sense of
Pringsheim.
Theorem G2 (Goginava [10]). Let α+ β ≥ 1, α, β > 0. Then there exists a continuous function
f0 ∈ PBV such that the Cesáro (C;−α,−β) means σ−α,−βn,n f0 (0, 0 ) of the double Walsh – Fourier
series of f0 diverges.
In this paper we consider the convergence of Cesáro means of negative order of double Walsh –
Fourier series of functions from the classes C
({
i1−α
}
,
{
i1−β
})
V
(
I2
)
(see Theorem 1) . We also
consider the following problem: Let α, β ∈ (0, 1) , α + β < 1. Under what conditions on the
sequence Λ = {λn} the double Walsh – Fourier series of the function f ∈ PΛBV is (C;−α,−β)
summable. The solution is given in Theorem 2 bellow.
4. Main results. The main results of this paper are presented in the following propositions:
Theorem 1. Let f ∈ C
({
i1−α
}
,
{
i1−β
})
V
(
I2
)
, α, β ∈ (0, 1). Then (C,−α,−β)-means of
double Walsh – Fourier series converges to f (x, y), if f is continuous at (x, y).
Theorem 2. Let Λ = {λn : n ≥ 1} , α+β < 1, α, β > 0,
λn
n1−(α+β)
↓ 0 and f ∈ PΛBV
(
I2
)
.
a) If
∞∑
n=1
λn
n2−(α+β)
<∞,
then (C;−α,−β)-means of double Walsh – Fourier series converges to f (x, y), if f is continuous at
(x, y).
b) If
∞∑
n=1
λn
n2−(α+β)
=∞,
then there exists a continuous function f ∈ PΛBV
(
I2
)
for which σ−α,−β2n,2n f (0, 0) diverges.
Corollary 1. Let α, β ∈ (0, 1) , α+ β < 1.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 495
a) If f ∈ P
{
n1−(α+β)
log1+ε (n+ 1)
}
BV (I2) for some ε > 0, then the double Walsh – Fourier series
of the function f is (C;−α,−β) summable to f (x, y), if f is continuous at (x, y).
b) There exists a continuous function f ∈ P
{
n1−(α+β)
log (n+ 1)
}
BV (I2) such that σ−α,−β2n,2n f (0, 0)
diverges.
Corollary 2. Let α, β ∈ (0, 1) , α + β < 1 and f ∈ PBV
(
I2
)
. Then the double Walsh –
Fourier series of the function f is (C;−α,−β) summable to f (x, y) , if f is continuous at (x, y) .
5. Auxiliary results.
Lemma 1. Let α ∈ (0, 1) and n := 2n1 + 2n2 + . . .+ 2nr , n1 > n2 > . . . > nr ≥ 0. Then
n∑
j=1
A−α−1
n−j Dj (x) =
r−1∑
l=1
(
l−1∏
k=1
w2nk (x)
)
D2nl (x)A−α
n(l−1)−1
−
−
r∑
l=1
(
l−1∏
k=1
w2nk (x)
)
w2nl−1 (x)
2nl−1∑
j=0
A−α−1
n(l)+j
Dj (x) .
Proof. Set
n(k) := 2nk+1 + 2nk+2 + . . .+ 2nr , nk+1 > nk+2 > . . . > nr ≥ 0, n(0) : = n.
Then from (3) and (7) can write
n∑
j=1
A−α−1
n−j Dj (x) =
2n1∑
j=1
A−α−1
n−j Dj (x) +
n(1)∑
j=1
A−α−1
n(1)−jDj+2n1 (x) =
=
2n1∑
j=1
A−α−1
n−j Dj (x) +D2n1 (x)A−α
n(1)−1
+ w2n1 (x)
n(1)∑
j=1
A−α−1
n(1)−jDj (x) .
Iterating this equality gives
n∑
j=1
A−α−1
n−j Dj (x) =
=
r∑
l=1
(
l−1∏
k=1
w2nk (x)
)
2nl∑
j=1
A−α−1
n(l−1)−jDj (x) +
r−1∑
l=1
(
l−1∏
k=1
w2nk (x)
)
A−α
n(l)−1
D2nl (x) . (9)
Since
D2n−l (x) = D2n (x)− w2n−1 (x)Dl (x) , l = 0, 1, . . . , 2n − 1,
we can write
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
496 U. GOGINAVA
2nl∑
j=1
A−α−1
n(l−1)−jDj (x) =
2nl∑
j=1
A−α−1
n(l)+2nl−jDj (x) =
2nl−1∑
j=0
A−α−1
n(l)+j
D2nl−j (x) =
= D2nl (x)
2nl−1∑
j=0
A−α−1
n(l)+j
− w2nl−1 (x)
2nl−1∑
j=0
A−α−1
n(l)+j
Dj (x) , l = 1, 2, ..., r. (10)
Combining (9) and (10) we obtain
n∑
j=1
A−α−1
n−j Dj (x) =
r∑
l=1
(
l−1∏
k=1
w2nk (x)
)
D2nl (x)A−α
n(l−1)−
−
r∑
l=1
(
l−1∏
k=1
w2nk (x)
)
w2nl−1 (x)
2nl−1∑
j=0
A−α−1
n(l)+j
Dj (x)−
−
(
r−1∏
k=1
w2nk (x)
)
D2nr (x)A−α
n(r−1)−1
=
r−1∑
l=1
(
l−1∏
k=1
w2nk (x)
)
D2nl (x)A−α
n(l−1)−1
−
−
r∑
l=1
(
l−1∏
k=1
w2nk (x)
)
w2nl−1 (x)
2nl−1∑
j=0
A−α−1
n(l)+j
Dj (x) .
Lemma 1 is proved.
Lemma 2. Let α ∈ (0, 1). Then
∣∣K−αn (x)
∣∣ ≤ c (α)
A−αn−1
|n|∑
l=0
2−lαD2l (x) .
Proof. From Lemma 1 we can write∣∣∣∣∣∣
n∑
j=1
A−α−1
n−j Dj (x)
∣∣∣∣∣∣ ≤
r∑
l=1
D2nl (x)A−α
n(l−1)+
+
r∑
k=1
2nk−1∑
j=1
∣∣∣A−α−1
n(k)+j
∣∣∣ |Dj (x)| : = B1 +B2. (11)
From (8) we have
B1 ≤ c (α)
|n|∑
l=0
2−lαD2l (x) . (12)
For B2 we can write
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ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 497
B2 =
r∑
k=1
nk∑
m=1
2m−1∑
j=2m−1
∣∣∣A−α−1
n(k)+j
∣∣∣ |Dj (x)| =
=
r∑
k=1
nk+1∑
m=1
2m−1∑
j=2m−1
∣∣∣A−α−1
n(k)+j
∣∣∣ |Dj (x)|+
r∑
k=1
nk∑
m=nk+1+1
2m−1∑
j=2m−1
∣∣∣A−α−1
n(k)+j
∣∣∣ |Dj (x)| .
From (4) and (8) we have
B2 ≤ c (α)
r∑
k=1
2nk+1(−α−1)
nk+1∑
m=1
2m
m∑
l=0
D2l (x) +
r∑
k=1
nk∑
m=nk+1+1
2m(−α−1)2m
m∑
l=0
D2l (x)
≤
≤ c (α)
n1∑
k=1
2−αk
k∑
l=0
D2l (x) ≤ c (α)
n1∑
l=0
2−αlD2l (x) . (13)
Combining (11) – (13) we complete the proof of Lemma 2.
Corollary 3. Let α ∈ (0, 1). Then
∣∣K−αn (x)
∣∣ ≤ cmin
{
1
A−αn−1
1
x1−α , n
}
.
Theorem B (Bakhvalov). Let Λi : =
{
λin : n ≥ 1
}
and Γi =
{
γin : n ≥ 1
}
such that γin =
= o
(
λin
)
, i = 1, 2. Then (
Γ1,Γ2
)
BV
(
I2
)
⊂ C
(
Λ1,Λ2
)
V
(
I2
)
.
Theorem 3. Let Λ = {λn : n ≥ 1} , α+ β < 1, α, β > 0. If
λn
n1−(α+β)
↓ 0 and
∞∑
n=1
λn
n2−(α+β)
<∞,
then there exists a sequence Γi =
{
γin : n ≥ 1
}
, i = 1, 2, such that γ1
n = o
(
n1−α) , γ2
n = o
(
n1−β)
and PΛBV
(
I2
)
⊂
(
Γ1,Γ2
)
BV
(
I2
)
.
Proof. By definition it is enough to prove that there exists a sequence Γi =
{
γin : n ≥ 1
}
,
i = 1, 2, with γ1
n = o
(
n1−α) , γ2
n = o
(
n1−β) such that for any f ∈ PΛBV
(
I2
)
Γ1V1
(
f ; I2
)
+ Γ2V2
(
f ; I2
)
+
(
Γ1,Γ2
)
V1,2
(
f ; I2
)
<∞.
Let the sequence {An : n ≥ 1} be such that
An ↑ ∞,
λnAn
n1−(α+β)
↓ 0,
∞∑
n=1
λnA
2
n
n2−(α+β)
<∞.
We set
Γ1 : =
{
γ1
n : =
n1−α
An
: n ≥ 1
}
, Γ2 : =
{
γ2
n : =
n1−β
An
: n ≥ 1
}
.
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We can write∑
i,j
|f (∆i, Jj)|
γ1
i γ
2
j
=
∑
i≤j
|f (∆i, Jj)|
γ1
i γ
2
j
+
∑
i>j
|f (∆i, Jj)|
γ1
i γ
2
j
: = F1 + F2. (14)
From the condition of the Theorem 3 we have
F1 ≤
∞∑
i=1
1
γ1
i
∞∑
j=i
|f (∆i, Jj)|
γ2
j
=
∞∑
i=1
Ai
i1−α
∞∑
j=i
|f (∆i, Jj)|
j1−β Aj ≤
≤ 2
∞∑
i=1
Ai
i1−α
sup
x
∞∑
j=i
|f (x, Jj)|
j1−β Aj = 2
∞∑
i=1
Ai
i1−α
sup
x
∞∑
j=i
|f (x, Jj)|
λj
λjAj
j1−β ≤
≤ 2ΛV2
(
f ; I2
) ∞∑
i=1
λiA
2
i
i2−(α+β)
<∞. (15)
Analogously, we can prove that
F2 <∞. (16)
Combining (14) – (16) we complete the proof of Theorem 3.
Theorem DF (Daly, Fridli [3]). Let n, N ∈ N and 1 < q ≤ 2. Then for any real numbers ck,
1 ≤ k ≤ 2n, we have
1∫
2−N
∣∣∣∣∣
2n∑
k=1
ckDk (x)
∣∣∣∣∣ dx ≤ c2N(1−1/q)
(
2n∑
k=1
|ck|q
)1/q
.
6. Proofs of main results. Proof of Theorem 1. It is easy to show that
σ−α,−βn,m f (x, y)− f (x, y) =
=
1
A−αn−1
1
A−βm−1
∫
I2
n∑
i=1
m∑
j=1
A−α−1
n−i A−β−1
m−j Di (s)Dj (t) ∆f (x, y, s, t) dsdt =
=
∫
IN−1×IM−1
+
∫
IN−1×IM−1
+
∫
IN−1×IM−1
+
∫
IN−1×IM−1
×
×
1
A−αn−1
1
A−βm−1
n∑
i=1
m∑
j=1
A−α−1
n−i A−β−1
m−j Di (s)Dj (t) ∆f (x, y, s, t)
: =
: = J1 + J2 + J3 + J4, (17)
where
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ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 499
∆f (x, y, s, t) : = f (xu s, y u t)− f (x, y) .
From the condition of the Theorem 1 and Corollary 3 we conclude that
|J1| ≤ c (α, β)nm
∫
IN−1×IM−1
|∆f (x, y, s, t)| dsdt = o (1) (18)
as n,m→∞.
For J2 we can write
|J2| ≤
c (β)m
A−αn−1
∫
IM−1
∣∣∣∣∣∣∣
∫
IN−1
2N−1∑
i=1
A−α−1
n−i Di (s) ∆f (x, y, s, t) ds
∣∣∣∣∣∣∣ dt+
+
c (β)m
A−αn−1
∫
IM−1
∣∣∣∣∣∣∣
∫
IN−1
n∑
i=2N−1+1
A−α−1
n−i Di (s) ∆f (x, y, s, t) ds
∣∣∣∣∣∣∣ dt : =
: = J21 + J22. (19)
From Theorem DF we obtain
|J21| ≤
c (β)
A−αn−1
N−1∑
l=0
∫
IM−1
∣∣∣∣∣∣∣
∫
Il\Il+1
2N−1∑
i=1
A−α−1
n−i Di (s) ∆ (x, y, s, t) ds
∣∣∣∣∣∣∣ dt ≤
≤ c (β)m
A−αn−1
N−1∑
l=0
ω (f ; IM−1 (x)× Il (y))×
∫
Il\Il+1
∣∣∣∣∣∣
2N−1∑
i=1
A−α−1
n−i Di (s)
∣∣∣∣∣∣ ds ≤
≤ c (α, β)
N−1∑
l=0
2(l−N)/2ω (f ; IM−1 (x)× Il (y)) =
= c (α, β)
∑
l≤N/2
+
∑
N/2<l<N
2(l−N)/2ω (f ; IM−1 (x)× Il (y)) ≤
≤ c (α, β, f)
{
2−N/4 + ω
(
f ; IM−1 (x)× I[N/2] (y)
)}
=
= o (1) as n, m→∞. (20)
For J22 we can write
|J22| ≤ c (β)m
A−αn−1
∫
IM−1
∣∣∣∣∣∣∣
∫
IN−1
2N∑
i=2N−1+1
A−α−1
n−i Di (s) ∆f (x, y, s, t) ds
∣∣∣∣∣∣∣ dt+
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500 U. GOGINAVA
+
c (β)m
A−αn−1
∫
IM−1
∣∣∣∣∣∣∣
∫
IN−1
n∑
i=2N+1
A−α−1
n−i Di (s) ∆f (x, y, s, t) ds
∣∣∣∣∣∣∣ dt =
= J1
22 + J2
22. (21)
From (2) we obtain
J1
22 =
c (β)m
A−αn−1
∫
IM−1
∣∣∣∣∣∣∣
∫
IN−1
2N−1∑
i=1
A−α−1
n−i−2N−1Di (s)w2N−1 (s) ∆f (x, y, s, t) ds
∣∣∣∣∣∣∣ dt =
=
c (β)m
A−αn−1
∫
IM−1
∣∣∣∣∣
2N−1−1∑
l=1
2N−1∑
i=1
A−α−1
n−i−2N−1Di
(
l
2N−1
)
×
×
∫
IlN−1
w2N−1 (s) ∆f (x, y, s, t) ds
∣∣∣∣∣dt. (22)
Since ( see [12])∫
IlN−1
w2N−1 (s) ∆f (x, y, s, t) ds =
∫
I2lN
∆N−1
0 f (xu s, y u t)1 ds
and
2N−1∑
i=1
A−α−1
n−i−2N−1Di (u) =
n−2N−1∑
i=1
A−α−1
n−i−2N−1Di (u)−
n−2N∑
i=1
A−α−1
n−i−2N
Di (u) (23)
from (8), (22) and Corollary 3 we can write
∣∣J1
22
∣∣ ≤ c (α, β)mn1−α
n−α
∫
IM−1×IN
2N−1−1∑
l=1
1
l1−α
∣∣∣∆N−1
l f (xu s, y u t)1
∣∣∣ dsdt. (24)
Set
µ (n,m) : =
[
min
{
N,
(
s (n,m)−1
)}]
,
where
s (n,m) := sup
0<s<(N+1)2−N ,0<t<2−M+1
|∆f (x, y, s, t)| .
Then from the condition of Theorem 1 and (24) we can write
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ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 501
∣∣J1
22
∣∣ ≤ c (α, β)nm
∫
IM−1×IN
µ(n,m)∑
l=1
1
l1−α
∣∣∣∆N−1
l f (xu s, y u t)1
∣∣∣ dsdt+
+c (α, β)nm
∫
IM−1×IN
2N−1−1∑
l=µ(n,m)+1
1
l1−α
∣∣∣∆N−1
l f (xu s, y u t)1
∣∣∣ dsdt ≤
≤ c (α, β)
{
s (n,m) (µ (n, n))α +
{
(i+ µ (n,m))1−α
}
V1
(
f ; I2
)}
≤
≤ c (α, β, f)
{
(s (n,m))1−α +
{
(i+ µ (n,m))1−α
}
V1
(
f ; I2
)}
=
= o (1) as n,m→∞. (25)
Analogously, we can prove that
J2
22 = o (1) as n,m→∞. (26)
Combining (21), (25) and (26) we obtain that
J22 = o (1) as n,m→∞. (27)
From (19), (20) and (27) we conclude that
J2 = o (1) as n,m→∞. (28)
Analogously, we can prove that
J3 = o (1) as n,m→∞. (29)
For J4, we can write
J4 =
1
A−αn−1
1
A−βm−1
∫
IN−1×IM−1
∑
(i,j)≤(2N−1,2M−1)
A−α−1
n−i A−β−1
m−j ×
×Di (s)Dj (t) ∆f (x, y, s, t) dsdt+
+
1
A−αn−1
1
A−βm−1
∫
IN−1×IM−1
∑
(i,j)
(2N−1,2M−1)
A−α−1
n−i A−β−1
m−j ×
×Di (s)Dj (t) ∆f (x, y, s, t) dsdt = J41 + J42. (30)
From Theorem DF we obtain
|J41| ≤
1
A−αn−1
1
A−βm−1
N−2∑
q=0
M−2∑
l=0
∣∣∣∣∣
∫
Iq\Iq+1
∫
Il\Il+1
2N−1∑
i=1
2M−1∑
j=1
A−α−1
n−i A−β−1
m−j ×
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×Di (s)Dj (t) ∆f (x, y, s, t) dsdt
∣∣∣∣∣ ≤ c (α, β)nαmβ
N−2∑
q=0
M−2∑
l=0
ω (f ; Iq (x)× Il (y))×
×
∫
Iq\Iq+1
∣∣∣∣∣∣
2N−1∑
i=1
A−α−1
n−i Di (s)
∣∣∣∣∣∣ ds
∫
Il\Il+1
∣∣∣∣∣∣
2M−1∑
j=1
A−β−1
m−j Dj (t)
∣∣∣∣∣∣ dt ≤
≤ c (α, β)
N−2∑
q=0
M−2∑
l=0
ω (f ; Iq (x)× Il (y)) 2(q−N)/22(l−M)/2 ≤
≤ c (α, β)
∑
0≤q<N/2
∑
0≤l<M/2
+
∑
0≤q<N/2
∑
M/2≤l<M
+
∑
N/2≤q<N
∑
0≤l<M/2
+
+
∑
N/2≤q<N
∑
M/2≤l<M
ω (f ; Iq (x)× Il (y)) 2(q−N)/22(l−M)/2 ≤
≤ c (α, β, f)
{
1
2(N+M)/4
+
1
2N/4
+
1
2M/4
+ ω
(
f ; I[N/2] (x)× I[M/2] (y)
)}
=
= o (1) as n,m→∞. (31)
Let i ≤ 2N−1 and 2M−1 < j ≤ 2M . Then we can write
J42 =
1
A−αn−1
1
A−βm−1
∫
IN−1
2N−1∑
i=1
A−α−1
n−i Di (s)×
×
( ∫
IM−1
2M−1∑
j=1
A−β−1
m−j−2M−1Dj (t)w2M−1 (t) ∆f (x, y, s, t) dt
)
ds =
=
1
A−αn−1
1
A−βm−1
∫
IN−1
2N−1∑
i=1
A−α−1
n−i Di (s)
2M−1−1∑
l=1
2M−1∑
j=1
A−β−1
m−j−2M−1Dj
(
l
2M−1
)
×
×
∫
I2lM
∆M−1
0 f (xu s, y u t)2 dt
ds.
Consequently, from Corollary 3 and (23) we obtain
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|J42| ≤
c (β)m
A−αn−1
2−[N/2]∫
2−N+1
∣∣∣∣∣∣
2N−1∑
i=1
A−α−1
n−i Di (s)
∣∣∣∣∣∣
∫
IM
2M−1−1∑
l=1
∆M−1
l f (xu s, y u t)2
l1−β
dt
ds+
+
c (β)m
A−αn−1
1∫
2−[N/2]
∣∣∣∣∣∣
2N−1∑
i=1
A−α−1
n−i Di (s)
∣∣∣∣∣∣
∫
IM
2M−1−1∑
l=1
∆M−1
l f (xu s, y u t)2
l1−β
dt
ds =
= J1
42 + J2
42. (32)
Set
r (n,m) : = sup
0<s<2−N/2,0<t<(2M+1)2−M
|∆f (x, y, s, t)|
and
θ (n,m) : =
[
min
{
M, r (n,m)−1
}]
.
Then applying Theorem DF for J1
42 we have
J1
42 ≤ c (β)m
A−αn−1
2−[N/2]∫
2−N+1
∣∣∣∣∣∣
2N−1∑
i=1
A−α−1
n−i Di (s)
∣∣∣∣∣∣
∫
IM
θ(n,m)∑
l=1
∆M−1
l f (xu s, y u t)2
l1−β
dt
ds+
+
c (β)m
A−αn−1
2−[N/2]∫
2−N+1
∣∣∣∣∣∣
2N−1∑
i=1
A−α−1
n−i Di (s)
∣∣∣∣∣∣
∫
IM
2M−1−1∑
l=θ(n,m)
∆M−1
l f (xu s, y u t)2
l1−β
dt
ds ≤
≤ c (α, β)
{
r (n,m) θβ (n,m) +
{
(l + θ (n,m))1−β V2
(
f ; I2
)}}
≤
≤ c (α, β)
{
r1−β (n,m) +
{
(l + θ (n,m))1−β V2
(
f ; I2
)}}
=
= o (1) as n,m→∞, (33)
J2
42 ≤
c (α, β)
{
i1−β
}
V2
(
f ; I2
)
2N/4
= o (1) as n,m→∞. (34)
Combining (32), (33) and (34) we conclude that
J42 = o (1) as n,m→∞. (35)
Analogously, we can prove that (35) holds in the cases when
(i, j) ∈
{
(i, j) : 0 ≤ i ≤ 2N−1, 2M < j ≤ m
}⋃
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504 U. GOGINAVA⋃{
(i, j) : 2N−1 < i ≤ 2N , 0 ≤ j ≤ 2M−1
}⋃{
(i, j) : 2N < i ≤ n, 0 ≤ j ≤ 2M−1
}
.
Let 2N−1 < i ≤ 2N and 2M < j ≤ m. Then we can write
J42 =
1
A−αn−1
1
A−βm−1
2N−1−1∑
k=1
2M−1∑
l=1
2N−1∑
i=1
m′∑
j=1
A−α−1
n−i−2N−1A
−β−1
m′−j Di
(
k
2N−1
)
Dj
(
l
2M
)
×
×
∫
I2kN ×I
2l
M+1
∆N−1,M
00 f (xu s, y u t) dsdt.
Set
p (n,m) : =
[
min
{
N,M, (ψ (n,m))−1/(2(α+β))
}]
,
where
ψ (n,m) : = sup
0<s<N+1
2N
, 0<t< 2M+1
2M+1
|∆f (x, y, s, t)| .
Then from the condition of the theorem we can write
|J42| ≤ c (α, β)nm
∫
IN×IM+1
2N−1−1∑
k=1
2M−1∑
l=1
1
k1−α
1
l1−β
∣∣∣∆N−1,M
kl f (xu s, y u t)
∣∣∣ dsdt ≤
≤ c (α, β)nm
∫
IN×IM+1
∑
(k,l)<(p(n,m),p(n,m))
1
k1−α
1
l1−β
∣∣∣∆N−1,M
kl f (xu s, y u t)
∣∣∣ dsdt+
+c (α, β)nm
∫
IN×IM+1
∑
(k,l)≮(p(n,m),p(n,m))
1
k1−α
1
l1−β
∣∣∣∆N−1,M
kl f (xu s, y u t)
∣∣∣ dsdt ≤
≤ c (α, β)
{
ψ (n,m) (p (n,m))α+β +
({
k1−α}{(l + p (n,m))1−β
})
V1,2
(
f, I2
)
+
+
({
(k + p (n,m))1−α
}{
l1−β
})
V1,2
(
f, I2
)}
=
= o (1) as n,m→∞. (36)
Analogously, we can prove that (36) holds in the cases when
(i, j) ∈
{
(i, j) : 2N−1 < i ≤ 2N , 2M−1 < j ≤ 2M
}⋃
⋃{
(i, j) : 2N < i ≤ n, 2M−1 < j ≤ 2M
}⋃{
(i, j) : 2N < i ≤ n, 2M < j ≤ m
}
.
From (30), (31), (35) and (36) we have
J4 = o (1) as n,m→∞. (37)
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ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 505
Combining (17), (18), (28), (29) and (37) we complete the proof of Theorem 1.
Proof of Theorem 2. The proof of the part a) of the Theorem 2 follows from Theorem B,
Theorems 1 and 3. Now, we prove the part b).
Consider the function ϕmN defined by
ϕmN (x) : =
2N+1x− 2j, x ∈
[
2j2−N−1, (2j + 1) 2−N−1
)
−
−
(
2N+1x− 2j − 2
)
, x ∈
[
(2j + 1) 2−N−1, (2j + 2) 2−N−1
)
,
j = 2m−1, . . . , 2m − 1.
Let
fN (x, y) : =
N∑
m=1
t2mϕ
m
N (x)ϕmN (y) sgn
(
K−α
2N
(x)
)
sgn
(
K−β
2N
(y)
)
,
where
tn : =
n∑
j=1
1
λj
−1
.
It is easy to show that fN ∈ PΛBV
(
I2
)
. Indeed, let y ∈
[
2m−N−1, 2m−N
)
for some m =
= 1, 2, ..., N. Then from the construction of the function fN we can write
∑
i
|fN (∆i, y)|
λi
≤ ct2m
2m∑
i=1
1
λi
≤ c <∞.
Consequently
ΛV1 (fN ) <∞. (38)
Analogously, we can prove that
ΛV2 (fN ) <∞. (39)
Combining (38) and (39) we conclude that fN ∈ PΛBV
(
I2
)
.
We can write
σ−α,−β
2N ,2N
fN (0, 0) =
∫
I2
fN (x, y)K−α
2N
(x)K−β
2N
(y) dxdy =
=
N∑
m=1
t2m
∫
[2m−N−1,2m−N )2
ϕmN (x)ϕmN (y)
∣∣K−α
2N
(x)
∣∣ ∣∣∣K−β2N
(y)
∣∣∣ dxdy ≥
≥ c
N∑
m=1
t2m
∫
[2m−N−1,2m−N )2
∣∣K−α
2N
(x)
∣∣ ∣∣∣K−β2N
(y)
∣∣∣ dxdy.
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506 U. GOGINAVA
Since [22] ∫
[2m−N−1,2m−N )
∣∣K−α
2N
(x)
∣∣ dx ≥ c (α) 2mα
we have ∣∣∣σ−α,−β2N ,2N
fN (0, 0)
∣∣∣ ≥ c (α, β)
N∑
m=1
t2m2m(α+β). (40)
Let λj : = γjj
1−(α+β). The from the condition of the Theorem 2 we obtain that γj ≥ γj+1.
Hence, we have
1
t2m
=
2m∑
i=1
1
λi
=
2m∑
i=1
1
i1−(α+β)γi
≤ c (α, β)
2m(α+β)
γ2m
,
t2m2m(α+β) ≥ c (α, β) γ2m .
Consequently, from (40) we have∣∣∣σ−α,−β2N ,2N
fN (0, 0)
∣∣∣ ≥ c (α, β)
N∑
m=1
γ2m = c (α, β)
N∑
m=1
λ2m
2m(1−(α+β))
→∞ as N →∞.
Applying the Banach – Steinhaus theorem, we obtain that there exists a continuous function f ∈
∈ PΛBV
(
I2
)
such that
sup
n
|σ−α,−β2n,2n f (0, 0) | =∞.
Theorem 2 is proved.
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Received 23.11.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
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| id | umjimathkievua-article-2591 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:25Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-25912020-03-18T19:30:15Z On the summability of double Walsh - Fourier series of functions of bounded generalized variation Про сумовнiсть подвiйних рядiв Уолша – Фур’є функцiй обмеженої узагальненої варiацiї Goginava, U. Гогінава, У. The convergence of Cesaro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated. Дослiджується збiжнiсть середнiх Чезаро вiд’ємного порядку вiд подвiйних рядiв Уолша – Фур’є функцiй обмеженої узагальненої варiацiї Institute of Mathematics, NAS of Ukraine 2012-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2591 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 4 (2012); 490-507 Український математичний журнал; Том 64 № 4 (2012); 490-507 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2591/1941 https://umj.imath.kiev.ua/index.php/umj/article/view/2591/1942 Copyright (c) 2012 Goginava U. |
| spellingShingle | Goginava, U. Гогінава, У. On the summability of double Walsh - Fourier series of functions of bounded generalized variation |
| title | On the summability of double Walsh - Fourier series of functions of bounded generalized variation |
| title_alt | Про сумовнiсть подвiйних рядiв Уолша – Фур’є функцiй обмеженої узагальненої варiацiї |
| title_full | On the summability of double Walsh - Fourier series of functions of bounded generalized variation |
| title_fullStr | On the summability of double Walsh - Fourier series of functions of bounded generalized variation |
| title_full_unstemmed | On the summability of double Walsh - Fourier series of functions of bounded generalized variation |
| title_short | On the summability of double Walsh - Fourier series of functions of bounded generalized variation |
| title_sort | on the summability of double walsh - fourier series of functions of bounded generalized variation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2591 |
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