On the summability of double Walsh–fourier series of functions of bounded generalized variation
The problem of convergence of the Cesàro means of negative order for double Walsh–Fourier series of functions of bounded generalized variation is investigated.
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irk-123456789-1641712020-02-23T19:23:17Z On the summability of double Walsh–fourier series of functions of bounded generalized variation Goginava, U. Статті The problem of convergence of the Cesàro means of negative order for 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 варiацi 2012 Article On the summability of double Walsh–fourier series of functions of bounded generalized variation / U. Goginava // Український математичний журнал. — 2012. — Т. 64, № 4. — С. 490-507. — Бібліогр.: 27 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164171 517.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Goginava, U. On the summability of double Walsh–fourier series of functions of bounded generalized variation Український математичний журнал |
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The problem of convergence of the Cesàro means of negative order for double Walsh–Fourier series of functions of bounded generalized variation is investigated. |
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Article |
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Goginava, U. |
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Goginava, U. |
| author_sort |
Goginava, U. |
| title |
On the summability of double Walsh–fourier series of functions of bounded generalized variation |
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On the summability of double Walsh–fourier series of functions of bounded generalized variation |
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On the summability of double Walsh–fourier series of functions of bounded generalized variation |
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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_sort |
on the summability of double walsh–fourier series of functions of bounded generalized variation |
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Інститут математики НАН України |
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2012 |
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Статті |
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| citation_txt |
On the summability of double Walsh–fourier series of functions of bounded generalized variation / U. Goginava // Український математичний журнал. — 2012. — Т. 64, № 4. — С. 490-507. — Бібліогр.: 27 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT goginavau onthesummabilityofdoublewalshfourierseriesoffunctionsofboundedgeneralizedvariation |
| first_indexed |
2025-07-14T16:41:48Z |
| last_indexed |
2025-07-14T16:41:48Z |
| _version_ |
1837641298166677504 |
| fulltext |
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
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
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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498 U. GOGINAVA
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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502 U. GOGINAVA
×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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ON THE SUMMABILITY OF DOUBLE WALSH – FOURIER SERIES OF FUNCTIONS . . . 503
|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.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 4
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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