GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 307-322
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
OF FUNCTIONS OF SEVERAL VARIABLES IN TERMS
OF FOURIER COEFFICIENTS
L. GOGOLADZE
Abstract. Inequalities are derived which enable one to estimate integral moduli of continuity of functions of several variables in terms
of Fourier coefficients.
In the paper inequalities are derived which give estimates of integral
moduli of continuity in terms of Fourier coefficients. These inequalities
generalize S. A. Telyakovskiı̆’s results [1] to the multi-dimansional case. For
simplicity and brevity we shall limit our discussion to the two-dimensional
case.
First we give some one-dimensional theorems for the series
∞
a0 X
+
ai cos ix,
2
i=1
∞
X
ai sin ix.
(1)
(2)
i=1
Let
∆ai = ai − ai+1 ,
[ 2i ]
X
∆ai−ν − ∆ai+ν
δai =
, i ≥ 2; δai = 0, i < 2,
ν
ν=1

−1 p

1 ≤ i ≤ m,
(im ) ,
αi,m (p) = 1,
i > m,


0,
i < 1.
(3)
(4)
In [1] and [2] the following theorems are proved.
1991 Mathematics Subject Classification. 42B05.
Key words and phrases. Integral modulus of continuity.
307
c 1999 Plenum Publishing Corporation
1072-947X/99/0700-0307$15.00/0 
308
L. GOGOLADZE
Theorem A1 . Let
lim ai = 0,
i→∞
∞
X
€
i=0

|∆ai | + |δai | < ∞.
Then series (1) converges for all x 6= 0 and for its sum f (x) we have
Zπ
−π
|f (x)|dx = O
’X
∞
i=0
“
€

|∆ai | + |δai | ,
“
’X
∞
 1‘
€

αi,m (p) |∆ai | + |δai | ,
=O
ωp f,
m L
i=0
(5)
€ 1
where ωp f, m
is an integral modulus of continuity of order p.
L
Theorem A2 . Let the conditions of Theorem A1 be fulfilled and
∞
X
|ai |
i=1
i
< ∞.
Then series (2) converges for all x ∈ (−π, π] and for its sum g(x) we have
the estimates
Zπ
π
2µ+1
|g(x)|dx =
µ
X
|ai |
i=1
i
+O
’X
∞
i=0
“
€

|∆ai | + |δai | ,
’X
“
∞
∞
 1 ‘ 2p X
€

|ai |
=
ωp g,
+O
αi,m (p) |∆ai | + |δai | .
m
π i=m i
i=0
(6)
Note that in formulas (5) and (6) the coefficients in the O-terms depend
only on p.
Now let a double sequence of numbers ai,j , i, j = 0, 1, . . . , be given and
set
∆1 ai,j = ai,j − ai+1,j , ∆2 ai,j = ai,j − ai,j+1 ,
∆1 ∆2 ai,j = ∆1 (∆2 ai,j ) = ∆2 (∆1 ai,j ),
∆21 ai,j = ∆1 (∆1 ai,j ),
δ1 ai,j
∆22 ai,j = ∆2 (∆2 ai,j ),
[ 2i ]
X
∆1 ai−ν,j − ∆1 ai+ν,j
, i ≥ 2; δ1 ai,j = 0, i < 2,
=
ν
ν=1
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
309
[ 2j ]
X
∆2 ai,j−µ − ∆2 ai,j+µ
=
, j ≥ 2; δ2 ai,j = 0, j < 2,
µ
µ=1
δ2 ai,j
δ1 ∆1 ai,j = δ1 (∆1 ai,j ) = ∆1 (δ1 ai,j ) = ∆1 δ1 ai,j ,
δ1 ∆2 ai,j = δ1 (∆2 ai,j ) = ∆2 (δ1 ai,j ) = ∆2 δ1 ai,j ,
δ2 ∆1 ai,j = δ2 (∆1 ai,j ) = ∆1 (δ2 ai,j ) = ∆1 δ2 ai,j ,
δ2 δ1 ai,j = δ2 (δ1 ai,j ) = δ1 (δ2 ai,j ) = δ1 δ2 ai,j .
Given a function f ∈ L(T 2 ), T = (−π, π], consider the expressions
(p,0)
∆t
f (x, y) =
p
X
(−1)µ
µ=0
q
X
’ “

€
p
f x + (p − 2µ)t, y ,
µ
’ “

q €
f x, y + (p − 2ν)s ,
ν
ν=0
€
€ (p,0)


(p,0)
(p,q)
∆(0,q)
f (x, y) = ∆s(0,q) ∆t f (x, y) ,
∆t,s f (x, y) = ∆t
s
Z
Π(p,0)
Œ
Œ∆t f (x, y)Œdx dy,
ω (p,0) (h, f ) = sup
∆(0,q)
f (x, y) =
s
(−1)ν
|t|≤h
ω
(0,q)
(η, f ) = sup
|s|≤η
T2
Z
T2
ω (p,q) (h, η, f ) = sup
Œ
Π(0,q)
Œ∆s f (x, y)Œdx dy,
Z
|t|≤h
2
|s|≤η T
Π(p,q)
Œ
Œ∆t,s f (x, y)Œdx dy.
Consider the series
∞ X
∞
X
λi,j ai,j cos ix cos jy,
(7)
i=0 j=0
where λ0,0 = 14 , λ0,j = λi,0 = 12 .
Theorem 1. Let
lim ai,j = 0,
i+j→∞
Ai,j = |∆1 ∆2 ai,j | + |δ1 ∆2 ai,j | + |∆1 δ2 ai,j | + |δ1 δ2 ai,j |,
∞ X
∞
X
i=0 j=0
Ai,j < ∞.
(8)
(9)
(90 )
310
L. GOGOLADZE
Then series (7) converges for all x 6= 0, y 6= 0 and for its sum f (x, y) we
have
“
’X
Z
∞
∞ X
|f (x, y)|dx dy = O
Ai,j ,
(10)
i=0 j=0
T2
Z ’
T
Z ’
sup
1
|s|≤ n
sup
1
|t|≤ m
T
sup
1
|t|≤ m
Z
|∆s(0,q) f (x, y)|dy
T
Z
(p,0)
|∆t f (x, y)|dx
T
Z ’
T
sup
1
|s|≤ n
=O
sup
1
|s|≤ n
Z ’
T
“
Z
T
dx = O
“
sup
αj,n (q)Ai,j = O(Bn ), (11)
i=0 j=0
dy = O
’X
∞ X
∞
αi,m (p)αj,n (q)Ai,j
(p,q)
|∆t,s f (x, y)|dx
T
“
“
αi,m (p)Ai,j = O(Cm ), (12)
i=0 j=0
i=0 j=0
1
|t|≤ m
“
“
(p,q)
|∆t,s f (x, y)|dy dx =
’X
∞
∞ X
Z
’X
∞ X
∞
= O(Dm,n ),
“
dy = O(Dm,n ).
(13)
(14)
Note that the coefficient in the O-term depends only on q in (11), only
on p in (12), and only on p and q in (13) and (14).
Proof. (9) and (90 ) imply
∞
∞ X
X
i=0 j=0
|∆1 ∆2 ai,j | < ∞.
(15)
This and (8) give for each j ≥ 0
∞
X
i=0
and
|∆1 ai,j | =
∞
∞ ŒX
∞ X
∞
ΠX
X
Œ
Œ
∆2 ∆1 ai,ν Œ ≤
|∆1 ∆2 ai,ν | < ∞
Œ
i=0
ν=j
lim
j→∞
(16)
i=0 ν=j
∞
X
i=0
|∆1 ai,j | = 0.
In a similar manner, for any i ≥ 0 we obtain
∞
X
j=0
|∆2 ai,j | < ∞
(160 )
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
311
and
lim
i→∞
∞
X
j=0
|∆2 ai,j | = 0.
(17)
Further, applying the Hardy transformation, we have
n
m X
X
λi,j ai,j cos ix cos jy =
i=0 j=0
+
m−1
X
m−1
X
X n−1
Di (x)Dj (y)∆1 ∆2 ai,j +
i=0 j=0
Di (x) cos ny∆1 ai,n +
n−1
X
Dj (y) cos mx∆2 am,j + Dm (x)Dn (y)am,n .
j=0
i=0
Now, using the property of Dirichlet kernels and (8), (15), (160 ), (17),
6 0. Moreover,
one can easily verify that series (7) converges for all x =
6 0, y =
(8) and (16) imply that the series
∞
a0,j X
+
ai,j cos ix = fj (x)
2
i=1
(18)
6 0.
converges for each j ≥ 0 and all x 6= 0, y =
As it is known (see, e.g., [3], p. 14) if a double series
∞
P
uij is convergent
i,j=0
to S and the series
∞
P
uij is convergent for each j, then the double series is
i=0
repeatedly convergent to S, i.e.,
∞
∞ ’X
X
uij
i=0
i=0
“
= S.
Therefore, series (7) will converge repeatedly and for its sum f (x, y) the
equality
∞
f0 (x) X
fj (x) cos jy
+
2
j=1
f (x, y) =
(19)
6 0
will be fulfilled. Hence, in turn, it follows that for each x =
lim fj (x) = 0
j→∞
ƒ
‚
and for each fixed t ∈ − n1 , n1 we have
(p,0)
∆t
(p,0)
f (x, y) =
∆t
∞
f0 (x) X (p,0)
+
∆t fj (x) cos jy
2
j=1
(20)
(21)
312
L. GOGOLADZE
for all (except the finite number of values of) x.
From (18) we obtain
∞
∆fj (x) =
δfj (x) =
∆2 a0,j X
+
∆2 ai,j cos ix,
2
i=1
∞
δ2 a0,j X
δ2 ai,j cos ix,
+
2
i=1
(22)
(23)
while (9) and (9’) imply
∞
X
€
i=0

|∆1 ∆2 ai,j | + |δ1 ∆2 ai,j | + |∆1 δ2 ai,j | + |δ1 δ2 ai,j | < ∞
(24)
for each j ≥ 0.
Moreover, for each j ≥ 0 (8) yields
lim ∆2 ai,j = 0,
i→∞
lim δ2 ai,j = 0.
i→∞
(25)
In view of the latter two relations Theorem A1 being applied to (22) and
(23) results
Z
|∆fj (x)|dx = O
T
Z
T
Z
T
|δfj (x)|dx = O
’X
∞
i=0
’X
∞
i=0
€
“

|∆1 ∆2 ai,j | + |δ1 ∆2 ai,j | ,
“
€

|∆1 δ2 ai,j | + |δ1 δ2 ai,j | ,
’X
∞
Œ
Œ
€
Œ∆t(p,0) (∆fj (x))Œdx = O
αi,m (p) |∆1 ∆2 ai,j | +
“

+ |δ1 ∆2 ai,j | ,
|t| ≤
’X
Z X
∞
∞ X
∞
€

€
|∆fj (x) + |δfj (x)| dx = O
|∆1 ∆2 ai,j | +
j=0
(28)
i=0
“

1
+ |δ1 δ2 ai,j | , |t| ≤ .
m
By summing (26) and (27) we obtain
T
(27)
i=0
1
,
m
’X
Z
∞
Π(p,0)
Œ
€
Œ∆t (δfj (x))Œdx = O
αi,m (p) |∆1 δ2 ai,j | +
T
(26)
i=0 j=0
(29)
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
313
“

+|δ1 ∆2 ai,j | + |∆1 δ2 ai,j | + |δ1 δ2 ai,j | .
Since the right-hand side of this inequality is finite by virtue of (9) and (90 ),
we have
∞
X

€
|∆fj (x)| + |δfj (x)| < ∞
j=0
for almost all x ∈ T . Thus for each fixed |t| ≤
1
m
we obtain
∞ 
X
Œ
Œ‘
Œ Œ
(p,0)
Œ∆(∆(p,0)
fj (x))Œ + Œδ(∆t fj (x))Œ < ∞
t
j=0
for almost all x.
Using the latter two relations and (20), by virtue of Theorem A1 we have
for series (19), (21)
Z
|f (x, y)|dy = O
T
sup
1
|s|≤ n
sup
1
|s|≤ n
Z
T
Z
T
=O
’X
∞
j=0
€
|∆fj (x)| + |δfj (x)|
“

,
’X
“
∞
Œ
Œ

€
Œ∆s(0,q) f (x, y)Œdy = O
αj,n (q) |∆fj (x)| + |δfj | ,
j=0
Œ
Œ
Œ∆s(0,q) (∆(p,0)
f (x, y))Œdy = sup
t
’X
∞
j=0
1
|s|≤ n
Z
T
Œ
Π(p,q)
Œ∆t,s f (x, y))Œdx dy =
“
Œ
Œ Œ
Œ‘
(p,0)
(p,0)
αj,n (q) Œ∆(∆t fj (x))Œ + Œδ(∆t fj (x))Œ .
Integrating these inequalities with respect to x and using (26)–(29), we
obtain the validity of (10), (11), and (13).
If instead of series (18) we consider the series
∞
ai,0 X
+
ai,j cos jy,
2
j=1
then by applying arguments similar to those above we shall obtain the
validity of (12), (14).
Theorem 2. Let the conditions of Theorem 1 be fulfilled and
∞ X
∞ ’
X
|ai,j |
i=1 j=1
ij
|∆1 ai,j | + |δ1 aij | |∆2 ai,j | + |δ2 ai,j |
+
+
j
i
“
< ∞.
(30)
314
L. GOGOLADZE
Then the series
∞ X
∞
X
ai,j sin ix sin jy
(31)
i=1 j=1
converges for all x ∈ T 2 and for its sum g(x, y) the following estimates are
valid:
Zπ
Zπ
π
π
2µ+1 2ν+1
+O
|g(x, y)|dx dy =
’X
µ X
∞
i=1 j=1
Zπ ’
π
2µ+1
sup
1
|s|≤ n
+O
+O
Z
1
2ν+1
T
’X
∞ X
∞
’X
µ X
∞
1
|t|≤ m
Z
αj,n (q)
T
i=m j=1
’X
∞ X
ν
i=1 j=1
+O
’X
∞ X
∞
i=1 j=1
“
|∆1 ai,j | + |δ1 ai,j |
j
“
Ai,j ,
“
dy =
|∆2 ai,j | + |δ2 ai,j |
i
“
“
+ O(Bn ) = O(En,µ ),
T
n
(32)
(33)
+
(34)
T
“
|∆1 ai,j | + |δ1 ai,j |
+
j
i=1 j=n
’X
“
∞
∞ X
|∆2 ai,j |+|δ2 ai,j |
αj,n (q)
+O
+O(Dm,n ) = O(Gm,n ),
i
i=m j=1
+O
+
∞
ν
2p X X |ai,j |
+
π i=m j=1 ij
“
|∆1 ai,j | + |δ1 ai,j |
+O
αi,m (p)
+ O(Cm ) = O(Fm,ν ),
j
i=1 j=1
“
Z ’
Z
∞ ∞
2p+q X X |ai,j |
(p,q)
sup
sup
+
|∆t,s g(x, y)|dy dx = 2
π i=m j=n ij
|t|≤ 1
|s|≤ 1
m
“
+
|∆2 ai,j | + |δ2 ai,j |
i
(p,0)
|∆t g(x, y)|dx
’X
∞ X
∞
“
’X
∞ X
ν
µ ∞
2q X X |ai,j |
dx =
+
π i=1 j=n ij
|∆1 ai,j | + |δ1 ai,j |
j
i=1 j=n
sup
+O
|∆2 ai,j | + |δ2 ai,j |
i
“
+O
ij
i=1 j=1
g(x, y)|dy
∆(0,q)
s
i=1 j=1
Zπ ’
µ X
ν
X
|ai,j |
’X
∞ X
∞
αi,m (p)
(35)
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
sup
1
|t|≤ n
Z ’
T
sup
1
|t|≤ m
“
(p,q)
|∆t,s g(x, y)|dx dy = O(Cm,n ).
Z
T
315
(36)
Note that the coefficient in the O-term depends only on q in (33), only
on p in (34) and only on p and q in (35) and (36).
Proof. The convergence of the series
∞
X
ai,j sin ix
i=1
and series (31) is proved in the same manner as that of series (18) and
(7). Denote their sums by gj (x) and g(x, y), respectively. The proof of the
equalities
gj (x) =
∞
X
(180 )
ai,j sin ix,
i=1
∞
X
g(x, y) =
(190 )
gj (x) sin jy
j=1
repeats that of equalities (18) and (19).
The validity of the equalities
(200 )
lim gj (x) = 0,
j→∞
(p,0)
∆t
∞
X
g(x, y) =
(p,0)
∆t
gj (x) sin jy,
j=1
∆gj (x) =
δgj (x) =
∞
X
|t| ≤
1
,
m
(210 )
(220 )
∆2 ai,j sin ix,
i=1
∞
X
(230 )
δ2 ai,j sin ix,
i=1
is obtained by using the same arguments as for equalities (20)–(23).
Inequality (30) implies that for each j ≥ 1
∞
X
|∆2 ai,j |
i=1
∞
X
i=1
i
< ∞,
|ai,j |
< ∞,
i
∞
X
|δ2 ai,j |
i=1
∞
X
i=1
i
< ∞,
|∆1 ai,j | < ∞,
∞
X
i=1
(37)
|δ1 ai,j | < ∞.
(38)
316
L. GOGOLADZE
Using (8), (24), (25), (37), and (38), by virtue of Theorem A2 we obtain
for series (180 ), (220 ), (230 )
Zπ
|gj (x)|dx =
π
2µ+1
Zπ
|∆gj (x)|dx =
Zπ
|δgj (x)|dx =
π
2µ+1
π
2µ+1
µ
X
|ai,j |
i=1
i
+O
i=1
µ
X
|∆2 ai,j |
i
i=1
µ
X
|δ2 ai,j |
i
i=1
’X
∞
+O
+O
“

€
|∆1 ai,j | + |δ1 ai,j | ,
’X
∞
€
i=1
’X
∞
i=1
(39)
“

|∆1 ∆2 ai,j |+|δ1 ∆2 ai,j | ,
(40)
“
€

|∆1 δ2 ai,j |+|δ1 δ2 ai,j | ,
(41)
’X
∞
∞
p X
Œ
Π(p,0)
€
|ai,j |
Œ∆t gj (x)Œdx = 2
+O
αi,m (p) |∆1 ai,j | +
π i=m i
i=1
T
“

1
+ |δ1 ai,j | , |t| ≤ ,
(42)
m
Z
∞
∞
p X
Π(p,0)
Œ
€X
€
|∆2 ai,j |
Œ∆t (∆gj (x))Œdx = 2
+O
αi,m (p) |∆1 ∆2 ai,j | +
π i=m
i
i=1
T
“

1
+ |δ1 ∆2 ai,j | , |t| ≤ ,
(43)
m
’X
Z
∞
∞
p X
Œ
Π(p,0)
€
|δ2 ai,j |
Œ∆t (δgj (x))Œdx = 2
+O
αi,m (p) |∆1 δ2 ai,j | +
π i=m
i
i=1
T
“

1
(44)
+ |δ1 δ2 ai,j | , |t| ≤ .
m
Z
After multiplying (39) by 1/j, passing to the limit in (39)–(41) as µ → ∞,
and then summing them with respect to the index j, we obtain
Z ’X
∞ X
∞
∞ 
‘“
X
|gj |
|ai,j |
+ |∆gj (x)| + |δgj (x)| dx =
+
j
ij
i=1 j=1
j=1
T
+
∞ X
∞
X
|∆2 ai,j | + |δ2 ai,j |
i=1 j=1
’X
∞ X
∞
+O
i=1 j=1
i
€
+O
’X
∞ X
∞
i=1 j=1
|∆1 ai,j | + |δ1 ai,j |
j
“
|∆1 ∆2 ai,j | + |δ1 δ2 ai,j | + |∆1 δ2 ai,j | + |δ1 δ2 ai,j |
+

“
.
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
317
Since by virtue of (9), (90 ), and (30) the right-hand side of this inequality
is finite, we have
∞ 
‘
X
|gj |
+ |∆gj (x)| + |δgj (x)| < ∞
j
j=1
for almost all x ∈ T . Hence for each fixed |t| ≤
∞ 
(p,0)
X
|∆
gj |
t
j
j=1
1
m
we have
Œ
ΠΠ(p,0)
Œ‘
(p,0)
+ Œ∆(∆t gj (x))Œ + Œδ(δt gj (x))Œ < ∞
for almost all x.
By virtue of Theorem A2 the latter two relations and (200 ) imply for
series (190 ) and (210 ) that
Zπ
π
2µ+1
|g(x, y)|dy =
ν
X
|gj |
j=1
j
+O
’X
∞
j=1
“

€
|∆gj (x)| + |δgj | ,
’X
“
∞
∞
q X
Œ
Œ
€

|gj |
Œ∆s(0,q) g(x, y)Œdy = 2
+O
αj,n (q) |∆gj | + |δgj | ,
1
π j=n j
|s|≤ n
j=1
T
Z
Z
Π(p,q)
Œ
Π(0,q) (p,0)
Œ
Œ∆t,s g(x, y)Œdy = sup
Œ∆s (∆t g(x, y))Œdy =
sup
sup
1
|s|≤ n
Z
T
1
|s|≤ n
T
’X
∞
∞
(p,0)
€
2 X |∆t gj (x)|
(p,0)
+O
αj,n (q) |∆(∆t gj (x))| +
=
π j=n
j
j=1
“

(p,0)
+ |δ(∆t gj (x)| .
q
After integrating these inequalities with respect to x and using (39)–(44),
we obtain the validity of (32), (33), (35).
If instead of series (180 ) we consider the series
∞ X
∞
X
ai,j sin jy,
i=1 j=1
then by a similar reasoning we shall obtain the validity of (34) and (36).
Corollary 1. Let the conditions of Theorem 1 be fulfilled. Then
1 ‘
1 ‘
ω (p,0)
, f = O(Cm ), ω (0,q) , f = O(Bn ),
m
n
‘

(p,q) 1 1
, , f = O(Dm,n ).
ω
m n
318
L. GOGOLADZE
Corollary 2. Let the conditions of Theorem 2 be fulfilled. Then
1 ‘
1 ‘
ω (p,0)
, g = O(Fm ), ω (0,q) , f = O(En ),
m
n
‘

(p,q) 1 1
, , f = O(Gm,n ).
ω
m n
where
En = lim En,µ , Fm = lim Fm,ν .
µ→∞
ν→∞
It is easy to verify that for each function ϕ ∈ L(T 2 )
“
’Z 
Z

‘
Π(p,0)
Œ ‘
(p,0) 1
Œ
Œ
,ϕ = O
∆t ϕ(x, y) dx dy ,
ω
sup
1
m
|t|≤ m
T
T
’Z 
“
Z
1 ‘
Π(0,q)
Œ ‘
(0,q)
Œ
Œ
ω
,ϕ = O
sup
∆s ϕ(x, y) dy dx ,
1
n
|s|≤ n
T
T
’
“
’
Z 
Z
Π(p,q)
Œ ‘
1 1 ‘
, , ϕ = O sup
sup Œ∆t,s ϕ(x, y)Œdy dx .
ω (p,q)
m n
|t|≤ 1
|s|≤ 1
m
T
n
(45)
(46)
(47)
T
These equalities and (11)–(13) imply the validity of Corollary 1. Corollary 2 is proved by a similar reasoning, but first we must pass to the limit
in (33) as µ → ∞ and in (34) as ν → ∞.
Corollary 3. Let equality (8) be fulfilled and
∞ X
∞
X
(i + 1)(j + 1)|∆21 ∆22 ai,j | < ∞.
i=0 j=0
Then
Z
|f (x, y)|dx dy = O
T2
Z 
T
sup
1
|s|≤ n
=O
Z 
T
Z
T
(i + 1)(j +
i=0 j=0
1
|s|≤ m
=O
Z
i=0 j=0
“
(i + 1)(j + 1)|∆21 ∆22 ai,j | ,
‘
f (x, y)|dy dx =
|∆(0,q)
s
’X
∞
∞ X
sup
’X
∞ X
∞
(p,0)
|∆t
T
’X
∞
∞ X
i=0 j=0
1)αj,n (q)|∆21 ∆22 ai,j |
‘
f (x, y)|dx dy =
(i + 1)(j +
(48)
“
= O(Hn ),
(49)
“
= O(Kn ),
(50)
1)αi,m (p)|∆21 ∆22 ai,j |
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
Z ’
sup
1
|t|≤ m
T
=O
sup
1
|s|≤ n
’X
∞
∞ X
sup
1
|s|≤ n
T
sup
1
|t|≤ m
“
(p,q)
T
Z
319
|∆t,s f (x, y)|dy dx =
(i + 1)(j +
i=0 j=0
Z ’
Z
“
1)αi,m (p)αj,n (q)|∆12 ∆22 ai,j |
(p,q)
|∆s,t f (x, y)|dx
T
“
dy = O(Lm,n ).
= O(Lm,n ), (51)
(52)
As one can easily verify, to prove Corollary 3 it suffices to estimate the
right-hand sides of relations (10)–(14) by the right-hand sides of (48)–(52),
respectively.
In [1] it is shown that
∞
X
i=0
∞
X
i=0
αi,m (p)|∆ai | = O
αi,m (p)|δai | = O
’X
∞
i=0
’X
∞
i=0
“
(i + 1)αi,m (p)|∆ ai | ,
2
(53)
“
(i + 1)αi,m (p)|∆2 ai |
(54)
are valid. In a similar manner one can prove
∞
X
i=0
∞
X
i=0
|∆ai | = O
|δai | = O
’X
∞
i=0
’X
∞
i=0
“
(i + 1)|∆ ai | ,
2
(55)
“
(i + 1)|∆2 ai | .
(56)
Applying (53) and (55) successively, we obtain
∞
∞ X
X
i=0 j=0
αj,n (q)|∆1 ∆2 ai,j | = O
=O
’X
∞ X
∞
i=0 j=0
“
’X
∞ X
∞
(j + 1)αj,n (q)|∆1 ∆22 ai,j | =
i=0 j=0
“
(i + 1)(j + 1)αj,n (q)|∆21 ∆22 ai,j | .
By a similar technique we derive the estimates
∞ X
∞
X
i=0 j=0
∞ X
∞
X
i=0 j=0
αj,n (q)|δ1 ∆2 ai,j | = O
αj,n (q)|∆1 δ2 ai,j | = O
’X
∞ X
∞
i=0 j=0
’X
∞ X
∞
i=0 j=0
“
(i + 1)(j + 1)αj,n (q)|∆21 ∆22 ai,j | ,
(i + 1)(j +
1)αj,n (q)|∆21 ∆22 ai,j |
“
,
320
L. GOGOLADZE
∞
∞ X
X
i=0 j=0
αj,n (q)|δ1 δ2 ai,j | = O
’X
∞ X
∞
i=0 j=0
“
(i + 1)(j + 1)αj,n (q)|∆21 ∆22 ai,j | .
The latter four relations enable us to estimate (see (9)) the right-hand side
of (11) by the right-hand side of (49). In the same manner one can prove
the validity of relations (48), (50)–(52).
Let

−1 p

1 ≤ i ≤ m,
(im ) ,
i
βi,m (p) = 1 + ln m ,
i > m,


0,
i < 1.
Corollary 4. Let (8) be fulfilled and
∞
∞ X
X
i=1 j=1
ij ln(i + 1) ln(j + 1)|∆21 ∆22 ai,j | < ∞.
Then
Z
|g(x, y)|dx dy = O
T2
Z 
T
sup
1
|s|≤ n
Z
T
=O
Z 
T
sup
1
|t|≤ m
sup
1
|t|≤ m
Z 
T
sup
1
|s|≤ n
Z 
T
(p,0)
|∆t
’X
∞
∞ X
sup
1
|s|≤ n
Z
T
i=1 j=1
sup
1
|t|≤ m
T
ij ln(i +
= O(Mn ),
“
= O(Nm ),
“
ijβi,m (p)βj,n (q)|∆12 ∆22 ai,j |
“
,
“
1)βi,m (p)|∆12 ∆22 ai,j |
‘
(p,q)
|∆t,s g(x, y)|dy dx =
’X
∞ X
∞
Z
1)βj,n (q)|∆21 ∆22 ai,j |
‘
g(x, y)|dx dy =
i=1 j=1
=O
ij ln(i +
i=1 j=1
T
=O
ij ln(i + 1) ln(j +
i=1 j=1
1)|∆21 ∆22 ai,j |
‘
g(x, y)|dy dx =
|∆(0,q)
s
’X
∞ X
∞
Z
’X
∞ X
∞
= O(Rm,n ),
‘
(p,q)
|∆t,s g(x, y)|dx dy = O(Rm,n ).
ON ESTIMATING INTEGRAL MODULI OF CONTINUITY
321
We have
∞
∞ ∞
∞ ν
∞
∞
X
X
1 ŒŒ X X 2 ŒŒ X 1 X X 2
|ai |
=
∆ aν Œ ≤
|∆ aν | ≤
Œ
i
i µ=i ν=µ
i ν=i µ=i
i=m
i=m
i=m
∞ X
ν
X
ν 2
|∆ aν | =
i
i
ν=m i=m
i=m ν=i
“
’X
∞
ν+1 2
=O
|∆ aν | .
ν ln
m
ν=m
≤
∞ X
∞
X
ν
|∆2 aν | =
(57)
The proof of Corollary 4 is obtained from Theorem 2 and (53)–(57) by
the reasoning similar to the proof of Corollary 3 by means of Theorem 1
and (53)–(56), but first one must pass to the limit in (33) as µ → ∞ and in
(34) as ν → ∞.
Corollary 5. Let the conditions of Corollary 3 be fulfilled. Then
1 ‘
1 ‘
, f = O(Km ), ω (0,q) , f = O(Hn ),
ω (p,0)
m
n

‘
(p,q) 1 1
ω
,
= O(Lm,n ).
m n
Corollary 6. Let the conditions of Corollary 4 be fulfilled. Then
1 ‘
1 ‘
, g = O(Mm ), ω (0,q) , g = O(Nn ),
ω (p,0)
m
n

‘
(p,q) 1 1
ω
, , g = O(Rm,n ).
m n
Corollaries 5 and 6 follow from Corollaries 3 and 4, respectively, if we use
(45), (46), (47). Concerning these corollaries, see [4], pp. 6–86.
One can also consider the series
∞ X
∞
X
i=0 j=1
λi,j cos ix sin jy,
∞ X
∞
X
λi,j sin ix cos jy.
i=1 j=0
The interested reader will readily know how to formulate and simultaneously prove the corresponding theorems for these series by the repeated
application of Theorems A1 and A2 .
Acknowledgment
This research was supported by the Grant of the Georgian Department
of Science and Technology (Contract No. 13–1).
322
L. GOGOLADZE
References
1. S. A. Telyakovskiı̆, The integrability of trigonometric series. Estimation of an integral modulus of continuity. (Russian) Mat. Sb. 91(133)
(1973), No. 4(8), 537–553.
2. S. A. Telyakovskiı̆, Conditions for the integrability of trigonometric
series and their application to the study of linear methods of summation of
Fourier series. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 28(1964), No.
6, 1209–1236.
3. V. G. Chelidze, Some methods of summation of double series and
double integrals. (Russian) Tbilisi University Press, Tbilisi, 1977.
4. T. Sh. Tevzadze, Special multiple trigonometric series. (Russian)
Tbilisi University Press, Tbilisi, 1998.
(Received 22.09.1997; revised 31.08.1998)
Author’s address:
Faculty of Mechanics and Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 380043
Georgia