Georgian Mathematical Journal
Volume 10 (2003), Number 2, 223–235
ON THE UNIFORM CONVERGENCE AND
L-CONVERGENCE OF DOUBLE FOURIER SERIES WITH
RESPECT TO THE WALSH–KACZMARZ SYSTEM
U. GOGINAVA
Abstract. In this paper we study the approximation by rectangular partial
sums of a double Fourier series with respect to the Walsh–Kaczmarz system
in the spaces C and L. From our results we obtain different criteria of the
uniform convergence and L-convergence of a double Fourier–Kaczmarz series.
2000 Mathematics Subject Classification: Primary 41A50; Secondary
42C10.
Key words and phrases: Walsh–Kaczmarz system, convergence in Lnorm, bounded variation in the sense of Hardy, uniform convergence.
1. Introduction
L. Zhizhiashvili ([13], Part. 2, Ch. 3) has established certain approximation
properties of rectangular partial sums of a double trigonometric Fourier series
in the spaces C and L. An analogous question for a double Fourier series with
respect to the Walsh–Paley system were treated in [2], [8].
We will study the approximation of a function f ∈ Lp in the norm of Lp for
p = 1 or p = ∞ by means of rectangular partial sums of a double Fourier series
with respect to the Walsh–Kaczmarz system; see Theorems 1, 2 and 3. From
these theorems one can obtain different criteria for the uniform convergence and
L1 -convergence of a double Fourier series with respect to the Walsh–Kaczmarz
system, in particular, establish a two-dimensional version of the Dini–Lipschitz
condition (see Corollaries 1 and 2).
Results of a somewhat different type can be obtained by using the variation
of a function.
Jordan [7] introduced a class of functions of bounded variation and, applying
it to the theory of Fourier series, he proved that if a continuous function has
bounded variation, then its Fourier series converges uniformly. In 1906 Hardy [6]
generalized the Jordan criterion to the double Fourier series and introduced, for
the function of two variables, the notion of bounded variation. He proved that if
the continuous function of two variables has bounded variation (in the sense of
Hardy), then its Fourier series converges uniformly in the sense of Pringsheim
(see, e.g., [13]). An analogous result for a double Walsh–Fourier series was
verified by Moricz [8]. The author proves in [3] that in Hardy’s theorem there
is no need to require the boundedness of nuxed variation (see V1,2 (f ) below), in
particular, it is proved that if f is continuous function and has bounded partial
variation, then its double trigonometric Fourier series converges uniformly on
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
224
U. GOGINAVA
[0, 2π]2 in the sense of Pringsheim. An analogous result for a double Walsh–
Fourier series is established in [4]. In this paper we study a similar question in
the case of the Walsh–Kaczmarz system (see Theorem 4).
2. Definitions and Notation
We denote the sets of all non-negative integers by N, the set of real numbers
by R and the set of dyadic rational numbers in the unit interval [0,1] by Q. In
particular, each element of Q has the form 2pn for some p, n ∈ N, 0 ≤ p ≤ 2n .
Let r0 (x) be the function
(
1
if x ∈ [0, 1/2)
r0 (x) =
, r0 (x + 1) = r0 (x) .
−1 if x ∈ [1/2, 1)
The Rademacher system is defined by rn (x) = r0 (2n x), n ≥ 1 and x ∈ [0, 1).
The Walsh system in the Paley enumeration {wn (x) : n ∈ N} is defined by
wn (x) =
m
Y
(rj (x))nj ,
j=0
where nj , j = 1, . . . , m, are the binary coefficients of n.
Now we recall the definition of the Walsh–Kaczmarz system {ψn (x) : n ∈ N} .
Set ψ0 (x) = 1, while for n ≥ 1 with the binary coefficients nk , k = 1, . . . , m,
set
m−1
Y
ψn (x) = rm (x)
(rm−j−1 (x))nj .
j=0
Let us consider the Dirichlet kernels
Dn (x) =
n−1
X
ωj (x) ,
Kn (x) =
j=0
n−1
X
ψj (x)
j=0
for the Walsh–Paley system and for the Walsh–Kaczmarz system, respectively.
We need the well-known equality for the Dirichlet kernel of the Walsh–Paley
system (see [5], p. 27)
(
2n if x ∈ [0, 1/2n ) ;
D2n (x) =
(1)
0 if x ∈ [1/2n , 1) .
The transformation τn for x ∈ [0, 1) is defined by
τn (x) =
n−1
X
−(k+1)
xn−k−1 2
k=0
where x =
∞
P
k=1
+
∞
X
xj 2−(j+1) ,
j=n
xk 2−(k+1) is the dyadic expansion of x (for x ∈ Q we choose the
expansion for which xk → 0 as k → ∞).
ON THE UNIFORM CONVERGENCE AND L-CONVERGENCE
225
We apply the transformation τn also for integers p ≥ 0 given by
τn (p) =
n−1
X
xn−j−1 2j ,
j=0
where
p=
n−1
X
xj 2j .
j=0
n
Given n ≥ 0 and 0 ≤ p < 2 , we set
In (p) = [p 2−n , (p + 1) 2−n ).
It is evident that the transformation τn (x) maps the segment In (p) on the
segment In (τn (p)) .
It is known [11] that
Kn (x) = D2k (x) + rk (x) Dm (τk (x))
for n = 2k + m, 0 ≤ m < 2k , (2)
and
2k
for x ∈ Ik (p) , p = 1, 2, . . . , 2k − 1.
(3)
τk (p)
We consider the double system {ψn (x) × ψm (y) : n, m ∈ N} on the unit
square I 2 = [0, 1) × [0, 1) . If f ∈ L (I 2 ) , then
|Dm (τk (x))| ≤
Z1 Z1
fˆ (n, m) =
f (x, y) ψn (x)ψm (y) dx dy
0
0
is the (n, m)-th Fourier coefficient of f.
The rectangular partial sums of double Fourier series with respect to the
Walsh–Kaczmarz system are defined by
SM,N (f ; x, y) =
M
−1 N
−1
X
X
fˆ (m, n) ψm (x)ψn (y).
m=0 n=0
2
As usual, denote by L (I ) the set of all measurable functions defined on I 2 ,
for which kf k1 , the integral of |f (x, y)| on I 2 , is finite. Furthermore, let C (I 2 )
be the set of all functions f : I 2 → R that are uniformly continuous from the
dyadic topology of I 2 to the usual topology of R with the norm (see [9], pp.
9–11)
¡
¡ ¢¢
kf kC = sup |f (x, y)|
f ∈ C I2 .
x,y∈I 2
The total modulus of continuity and the total integrated modulus of continuity are respectively defined by
°
ª
©°
ω (f ; δ1 , δ2 )C = sup °f (x ⊕ u, y ⊕ v) − f (x, y) °C : 0 ≤ u < δ1 , 0 ≤ v < δ2 ,
°
ª
©°
ω (f ; δ1 , δ2 ) = sup °f (x ⊕ u, y ⊕ v) − f (x, y) ° : 0 ≤ u < δ1 , 0 ≤ v < δ2 ,
1
1
226
U. GOGINAVA
where ⊕ denotes the dyadic addition (see, e.g., [5], [9]), while the partial moduli
of continuity and the partial integrated moduli of continuity are respectively
defined by
¡ ¢
ω1 (f ; δ)C = ω (f ; δ, 0)C and ω2 (f ; δ)C = ω (f ; 0, δ)C ,
f ∈ C I2 ,
¡ ¢
ω1 (f ; δ)1 = ω (f ; δ, 0)1 and ω2 (f ; δ)1 = ω (f ; 0, δ)1 ,
f ∈ L I2 .
We also use the notion of the mixed modulus of continuity, and the mixed
integrated modulus of continuity which are respectively defined as follows:
©°
ω1,2 (f ; δ1 , δ2 )C = sup °f (x ⊕ u, y ⊕ v) − f (x ⊕ u, y)
°
ª
¡ ¢
−f (x, y ⊕ v) + f (x, y) °C : 0 ≤ u < δ1 , 0 ≤ v < δ2 , f ∈ C I 2 ,
©°
ω1,2 (f ; δ1 , δ2 )1 = sup °f (x ⊕ u, y ⊕ v) − f (x ⊕ u, y)
°
ª
¡ ¢
−f (x, y ⊕ v) + f (x, y) ° : 0 ≤ u < δ1 , 0 ≤ v < δ2 , f ∈ L I 2 .
1
A function f : I 2 → R is said to be of bounded variation in the sense of
Hardy(f ∈ HBV (I 2 )) if there exists a constant K such that for any partition
∆1 : 0 ≤ x0 < x1 < x2 < · · · < xn ≤ 1,
∆2 : 0 ≤ y0 < y1 < y2 < · · · < ym ≤ 1,
we have
V1,2 (f ) = sup
∆1 ×∆2
n−1 m−1
X
X¯
¯f (xi , yj ) − f (xi+1 , yj )
i=0 j=0
¯
− f (xi , yj+1 ) + f (xi+1 , yj+1 ) ¯ ≤ K,
n−1
X
¯
¯
¯f (xi , y) − f (xi+1 , y) ¯ ≤ K,
V1 (f ) = sup sup
y
∆1
V2 (f ) = sup sup
x
∆2
i=0
m−1
X
¯
¯
¯f (x, yj ) − f (x, yj+1 ) ¯ ≤ K.
j=0
Definition 1 ([4]). We say that the function f : I 2 → R is of bounded
partial variation (f ∈ P BV (I 2 )) if V1 (f ) and V2 (f ) are finite.
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−1
∆m
, y − f x ⊕ (2j + 1) 2−m−1 , y ,
j f (x, y)1 = f x ⊕ 2j2
¡
¢
¡
¢
∆ni f (x, y)2 = f x, y ⊕ 2i2−n−1 − f x, y ⊕ (2i + 1) 2−n−1 ,
¢
¡ m
n
m
n
∆mn
ji f (x, y) = ∆i ∆j f (x, y)1 2 = ∆j (∆i f (x, y)2 )1
¢
¢
¡
¡
= f x ⊕ 2j2−m−1 , y ⊕ 2i2−n−1 − f x ⊕ (2j + 1) 2−m−1 , y ⊕ 2i2−n−1
¡
¢
− f x ⊕ 2j2−m−1 , y ⊕ (2i + 1) 2−n−1
¡
¢
+ f x ⊕ (2j + 1) 2−m−1 , y ⊕ (2i + 1) 2−n−1 .
ON THE UNIFORM CONVERGENCE AND L-CONVERGENCE
227
−1
m
Furthermore, set λm
for 1 ≤ j < 2m and
0 = 1 and λj = (τm (j))
Wm(1) (f ; x, y) =
m −1
2X
¯ m
¯
¯
¯
λm
∆
f
(x,
y)
j
j
1 ,
j=0
Wn(2)
(f ; x, y) =
n −1
2X
λni |∆ni f (x, y)2 | ,
i=0
Wmn (f ; x, y) =
m −1 2n −1
2X
X
j=0
¯
¯
n ¯ mn
¯
λm
j λi ∆ji f (x, y) .
i=0
2.1. Main Results.
Theorem 1. Let M, N be positive integers such that M = 2m +j, 0 ≤ j < 2m ,
and N = 2n + i, 0 ≤ i < 2n , for some integers m, n ≥ 0. If f ∈ C (I 2 ) , then
kSM,N (f ) − f kC
µ
¶
°
°
1 1
1°
1°
1
≤ ω f; m , n
+ °Wm(1) (f )°C + °Wn(2) (f )°C + kWmn (f )kC .
2 2 C 2
2
4
Theorem 2. Let M, N be positive integers such that M = 2m +j, 0 ≤ j < 2m ,
and N = 2n + i, 0 ≤ i < 2n , for some integers m, n ≥ 0. If f ∈ C (I 2 ); then
kSM,N (f ) − f kC
½ µ
¾
¶
µ
¶
µ
¶
1
1
1 1
≤ c ω1 f ; m
m + ω2 f ; n
n + ω1,2 f ; m , n
mn .1
2
2
2
2
C
C
C
Corollary 1. Let f ∈ C (I 2 ) and
¶
µ
1
m → 0 as
ω1 f ; m
2
C
µ
¶
1
ω2 f ; n
n → 0 as
2 C
µ
¶
1 1
mn → 0 as
ω1,2 f ; m , n
2 2 C
m → ∞,
n → ∞,
n, m → ∞;
then the double Fourier series with respect to Walsh–Kaczmarz system converges
uniformly on I 2 .
Theorem 3. Let M, N be positive integers such that M = 2m +j, 0 ≤ j < 2m ,
and N = 2n + i, 0 ≤ i < 2n , for some integers m, n ≥ 0. If f ∈ L (I 2 ); then
kSM,N (f ) − f k1
¶
µ
¶
µ
¶
¾
½ µ
1
1 1
1
≤ c ω1 f ; m m + ω2 f ; n n + ω1,2 f ; m , n mn .
2
2 1
2 2 1
1
1Here
places.
and below the constant c is an absolute constant and may be different in different
228
U. GOGINAVA
Corollary 2. Let f ∈ L (I 2 ) and
µ
¶
1
ω1 f ; m m → 0 as
2
µ
¶1
1
ω2 f ; n n → 0 as
2 1
µ
¶
1 1
ω1,2 f ; m , n mn → 0 as
2 2 1
m → ∞,
n → ∞,
n, m → ∞;
then the double Fourier series with respect to Walsh–Kaczmarz system converges
to f in L-norm.
Theorem 4. Let f ∈ C (I 2 ) ∩ P BV (I 2 ) . Then the double Fourier series with
respect to Walsh–Kaczmarz system converges uniformly on I 2 .
2.2. Proof of the Main Results.
Proof of Theorem 1. By (2) we write
SM,N (f ; x, y) − f (x, y)
Z1 Z1
=
[f (x ⊕ u, y ⊕ v) − f (x, y)] KM (u) KN (v) du dv
0
0
Z1
Z1
=
[f (x ⊕ u, y ⊕ v) − f (x, y)] D2m (u) D2n (v) du dv
0
0
Z1 Z1
+
[f (x ⊕ u, y ⊕ v) − f (x, y)] D2m (u) rn (v) Di (τn (v)) du dv
0
0
Z1
Z1
+
[f (x ⊕ u, y ⊕ v) − f (x, y)] D2n (v) rm (u) Dj (τm (u)) du dv
0
0
Z1
Z1
+
[f (x ⊕ u, y ⊕ v)−f (x, y)] rn (v) rm (u) Dj (τm (u)) Di (τn (v)) dudv
0
0
= I + II + III + IV.
(4)
We find by (1) that
m
n
° 1/2
°
Z 1/2
Z
°
°
n+m °
°
kIkC = 2
[f
(x
⊕
u,
y
⊕
v)
−
f
(x,
y)]
du
dv
°
°
0
0
µ
≤ ω f;
It is well-known (see, e.g., [5]) that
1 1
,
2m 2n
C
¶
.
C
(5)
ON THE UNIFORM CONVERGENCE AND L-CONVERGENCE
229
(
1 if u ∈ Im+1 (2j)
m ∈ N, 0 ≤ j < 2m ;
−1 if u ∈ Im+1 (2j + 1) ,
b) t = u ⊕ 2−m−1 is a one-to-one mapping of Im+1 (2j) onto Im+1 (2j + 1) .
Thus, by (1) and (a)–(b) ,
a)
w2m (u) =
Z
II = 2
Z1
m
[f (x ⊕ u, y ⊕ v) − f (x, y)] rn (v) Di (τn (v)) du dv
Im (0) 0
=2
m
Z
n −1
2X
r=0
³ Z
[f (x ⊕ u, y ⊕ v) − f (x, y)] rn (v) Di (τn (v)) dv du
Im (0)
X Z
=2
r=0
Z
In (r)
Z
³
2n −1
m
´
[f (x ⊕ u, y ⊕ v) − f (x, y)] Di (τn (v)) dv
Im (0)
In+1 (2r)
´
[f (x ⊕ u, y ⊕ v) − f (x, y)] Di (τn (v)) dv du
−
In+1 (2r+1)
=2
m
Z
n −1
2X
r=0
Z
£
¡
¢¤
f (x ⊕ u, y ⊕ v) − f x ⊕ u, y ⊕ v ⊕ 2−n−1
Im (0) In+1 (2r)
Z
× Di (τn (v)) du dv
Z
£
= 2m
¡
f (x ⊕ u, y ⊕ v) − f x ⊕ u, y ⊕ v ⊕ 2−n−1
¢¤
Im (0) In+1 (0)
× Di (τn (v)) du dv
Z
2n −1
+ 2m
X
Z
£ ¡
¢
f x ⊕ u, y ⊕ v ⊕ 2r2−n−1
Di (τn (r))
r=1
Im (0) In+1 (0)
¡
−f x ⊕ u, y ⊕ v ⊕ (2r + 1) 2−n−1
¢¤
du dv.
From (3) we have
Z
m
|II| ≤ 2
Z
|∆n0 f (x ⊕ u, y ⊕ v)2 | du dv
i
Im (0) In+1 (0)
Z
Z
+ 2m+n
Im (0) In+1 (0)
n −1
2X
r=1
1
|∆n f (x ⊕ u, y ⊕ v)2 | du dv.
τn (r) r
Consequently,
kIIkC ≤
°
1°
°Wn(2) (f )° .
C
2
(6)
230
U. GOGINAVA
The estimation of III is analogous to that of II and we have
kIIIkC ≤
°
1°
°Wm(1) (f )° .
C
2
(7)
Following a similar pattern for the case of II, by (a)–(b) we obtain
IV =
m −1 2n −1
2X
X
s=0
µ
Z
Di (τn (r)) Dj (τm (s))
Z
Z
Z
−
Z
Im+1 (2s) In+1 (2r)
+
Im+1 (2s+1) In+1 (2r)
=
2n −1
s=0
r=0
X X
Z
Z
−
r=0
2m −1
Z
¶
Im+1 (2s) In+1 (2r+1)
[f (x ⊕ u, y ⊕ v) − f (x, y)] du dv
Im+1 (2s+1) In+1 (2r+1)
Di (τn (r)) Dj (τm (s))
Z
Z
£ ¡
¢
f x ⊕ u ⊕ 2−m−1 , y ⊕ v ⊕ 2−n−1
×
Im+1 (2s) In+1 (2r)
¡
¢
− f x ⊕ u ⊕ 2−m−1 , y ⊕ v
¡
¢
¤
−f x ⊕ u, y ⊕ v ⊕ 2−n−1 + f (x ⊕ u, y ⊕ v) du dv
=
m −1 2n −1
2X
X
s=0
Z
r=0
Di (τn (r)) Dj (τm (s))
Z
£ ¡
¢
f x ⊕ u ⊕ (2s + 1) 2−m−1 , y ⊕ v ⊕ (2r + 1) 2−n−1
×
Im+1 (0) In+1 (0)
¡
¢
− f x ⊕ u ⊕ (2s + 1) 2−m−1 , y ⊕ v ⊕ 2r2−n−1
¡
¢
− f x ⊕ u ⊕ 2s2−m−1 , y ⊕ v ⊕ (2r + 1) 2−n−1
¡
¢¤
+f x ⊕ u ⊕ 2s2−m−1 , y ⊕ v ⊕ 2r2−n−1 du dv.
By (3) we obtain
Z
Z
|∆mn
00 f (x ⊕ u, y ⊕ v)| du dv
|IV | ≤ ij
Im+1 (0) In+1 (0)
Z
Z
n
+ j2
Im+1 (0) In+1 (0)
Z
Z
m
+ i2
Im+1 (0) In+1 (0)
n −1
2X
r=1
m −1
2X
s=1
1
|∆mn
f (x ⊕ u, y ⊕ v)| du dv
τn (r) 0r
1
|∆mn
f (x ⊕ u, y ⊕ v)| du dv
τm (s) s0
ON THE UNIFORM CONVERGENCE AND L-CONVERGENCE
Z
+2
Z
n −1 2m −1
2X
X
n+m
Im+1 (0) In+1 (0)
Z
r=1
s=1
231
1
1
|∆mn f (x ⊕ u, y ⊕ v)| du dv
τn (r) τm (s) sr
Z
≤ 2n+m
Wmn f (f ; x ⊕ u, y ⊕ v) du dv.
Im+1 (0) In+1 (0)
Consequently,
1
kWmn (f )kC .
4
Combining (4)–(8), we complete the proof of Theorem 1.
Proof of Theorem 2. Since
kIV kC ≤
n −1
2X
λnr
=1+
r=0
n −1
2X
r=1
and
(8)
¤
n
2X
−1
1
1
=1+
≤ cn
τn (r)
r
r=1
m −1
¶ 2X
µ
¶
1
1
m
(1)
λr ≤ cω1 f ; m m,
Wm (f ; x, y) ≤ ω1 f ; m
2
2
C r=0
n −1
µ
¶ 2X
µ
¶
1
1
(2)
n
Wn (f ; x, y) ≤ ω2 f ; n
λ ≤ cω2 f ; n n,
2 C i=0 i
2
n −1 2m −1
¶ 2X
µ
¶
µ
X
1 1
1 1
n m
λ λ ≤ cω1,2 f ; m , n nm,
Wmn (f ; x, y) ≤ ω1,2 f ; m , n
2 2 C r=0 i=0 r i
2 2
µ
the validity of Theorem 2 follows from Theorem 1.
¤
Calculations similar to those that were performed in the proofs of Theorems
1, 2 and the application of the Minkowski inequality yield the validity of Theorem 3.
Proof of Theorem 4. On the basis of Theorem 1 it suffices to show that
° (1)
°
°Wm (f )° → 0 as m → ∞,
° (2)
°C
°Wn (f )° → 0 as n → ∞,
C
kWmn (f )kC → 0
Let
as n, m → ∞.
©
ª
m − 1, 2r ≤ τ (j) < 2r+1 , r = 0, 1, . . . , m − 1.
Am
=
j
:
j
=
1,
2
m
r
Then it is evident that
r
|Am
r | < 2
and
m−1
[
r=0
m
Am
r = {1, 2, . . . , 2 − 1} .
(9)
(10)
232
U. GOGINAVA
By the condition of the theorem and (9), (10) we get
Wm(1)
(f ; x, y) =
|∆m
0 f
(x, y)1 | +
µ
≤ ω1 f ;
µ
≤ ω1
1
2m
1
f; m
2
µ
≤ ω1 f ;
1
2m
m−1
X
+
r=η(m)
m −1
2X
¶
+
C
j=1
¯
1 ¯¯ m
∆j f (x, y)1 ¯
τm (j)
m−1
X
X
r=0 j∈Am
r
¶
+
C
m−1
X
r=0
¶
¯
1 ¯¯ m
∆j f (x, y)1 ¯
τm (j)
¯
1 X ¯¯ m
¯
∆
f
(x,
y)
j
1
2r j∈Am
r
η(m)−1
+
C
X 1 X ¯
¯
¯∆m
¯
f
(x,
y)
j
1
r
2
r=0
j∈Am
r
¯
1 X ¯¯ m
¯
∆
f
(x,
y)
j
1
2r j∈Am
r
µ
¶
µ
¶ η(m)−1
m−1
X 1
X 1
1
1
m
≤ ω1 f ; m
+ ω1 f ; m
|A
|
+
V
(f
)
1
r
r
2
2
2r
C
C r=0 2
r=η(m)
½ µ
¶
¾
1
1
η (m) + η(m) → 0
≤ c ω1 f ; m
as m → ∞,
2
2
C
where
½ µ
¶
¾
¶
µ
1
1
1
1
η + η = ω1 f ; m
η (m) + η(m) ;
minm
ω1 f ; m
1≤η≤2 −1
2
2
2
2
C
C
consequently,
Analogously,
° (1)
°
°Wm (f )° → 0
C
as m → ∞.
(11)
° (2)
°
°Wn (f )° → 0
C
as n → ∞.
(12)
We write
Wmn (f ; x, y) = |∆mn
00 f (x, y)| +
n −1
2X
i=1
+
m −1
2X
j=1
¯
1 ¯¯ mn
∆j0 f (x, y)¯ +
τm (j)
m −1 2n −1
2X
X
j=1
i=1
1
|∆mn f (x, y)|
τn (i) 0i
¯
1
1 ¯¯ mn
∆ji f (x, y)¯
τn (i) τm (j)
= I + II + III + IV.
It is evident that
|∆mn
00 f
Since
(x, y)| ≤ ω1,2
µ
1 1
f; m , n
2 2
(13)
¶
→ 0 as n, m → ∞.
C
¯ ¯ m ¡
¯ ¯
¯ mn
¢¯
−n−1 ¯
¯ ¯
¯∆j0 f (x, y)¯ ≤ ¯∆m
,
j f (x, y)1 + ∆j f x, y ⊕ 2
1
(14)
ON THE UNIFORM CONVERGENCE AND L-CONVERGENCE
233
from (11) we get
°
°
kIIkC ≤ 2 °Wn(2) (f )°C → 0 as m → ∞.
(15)
kIIIkC → 0 as m → ∞.
(16)
Analogously,
From (9) and (10) we get
m−1
n−1
XX
IV =
X X
n
r=0 s=0 j∈Am
r i∈As
m−1
X
≤
r=0
Since
¯
1
1 ¯¯ mn
∆ji f (x, y)¯
τn (i) τm (j)
n−1
¯
1 X 1 X X ¯¯ mn
∆ji f (x, y)¯ .
r
s
2 s=0 2 j∈Am i∈An
r
(17)
s
X
X X¯
¯
m
¯∆mn
¯
f
(x,
y)
≤
2
|A
|
sup
|∆ni f (x, y)2 | ,
r
ji
n
j∈Am
r i∈As
x∈[0,1] i∈An
n
j∈Am
r i∈As
y∈[0,1] j∈Am
s
X X¯
X ¯
¯
¯
n
¯∆mn
¯
¯∆m
¯
ji f (x, y) ≤ 2 |As | sup
j f (x, y)1 ,
we have
r
µ X X
X X¯
¯
¯ mn
¯ ¢1/2
¯∆mn
¯
¯
¯
f
(x,
y)
=
∆
f
(x,
y)
ji
ji
n
j∈Am
r i∈As
n
j∈Am
r i∈As
µ X X
¶
·
¯ mn
¯ 1/2
n
¯∆ji f (x, y)¯
×
≤ 2 |Am
r | |As |
n
j∈Am
r i∈As
X
× sup
x∈[0,1] i∈An
|∆ni f
¸
X ¯
¯ 1/2
m
¯
¯
(x, y)2 | sup
∆j f (x, y)1
.
(18)
y∈[0,1] j∈Am
s
r
After substituting (18) in (17), we obtain by (9) and the condition of the
theorem that
m−1
X ¯
X 1 ³
¯ ´1/2
¯∆m
¯
sup
f
(x,
y)
IV ≤ 2
j
1
r/2
2
y∈[0,1]
r=0
j∈Am
r
×
s=0
=2
1 ³
n−1
X
2s/2
+
m−1
X
r=ϕ(m)
x∈[0,1] i∈An
|∆ni f
(x, y)2 |
´1/2
s
½ ϕ(m)−1
X
r=0
sup
X
1 ³
2r/2
X ¯
¯ ´1/2
m
¯
sup
∆j f (x, y)1 ¯
y∈[0,1] j∈Am
r
1 ³
2r/2
¾
X ¯
¯ ´1/2
m
¯∆j f (x, y) ¯
sup
1
y∈[0,1] j∈Am
r
234
U. GOGINAVA
×
½ ψ(n)−1
X
2s/2
s=0
+
n−1
X
s=ψ(n)
1 ³
sup
X
x∈[0,1] i∈An
|∆ni f
(x, y)2 |
s
1 ³
2r/2
sup
X
x∈[0,1] i∈An
|∆ni f
(x, y)2 |
´1/2
´1/2 ¾
s
½s µ
¶
¾
1
1
≤c
ω1 f ; m ϕ (m) + ϕ(m)/2
2
2
C
s
½
µ
¶
¾
1
1
×
ω2 f ; n ψ (n) + ψ(n)/2 → 0
2 C
2
as m, n → ∞,
(19)
where
¶
¾ s µ
¶
½s µ
1
1
1
1
ω1 f ; m
ϕ + ϕ/2 = ω1 f ; m
ϕ (m) + ϕ(m)/2
minm
1≤ϕ≤2 −1
2
2
2
2
C
C
and
½s
minn
1≤ψ≤2 −1
µ
ω2
1
f; n
2
¶
ψ+
C
1
s
¾
2ψ/2
=
µ
ω2
1
f; n
2
¶
ψ (n) +
C
1
2ψ(n)/2
.
Combining (13)–(16) and (19) we have
kWmn (f )kC → 0
as n, m → ∞.
From (11), (12) and (20) we complete the proof of Theorem 4.
(20)
¤
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ON THE UNIFORM CONVERGENCE AND L-CONVERGENCE
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(Received 28.04.2002)
Author’s address:
Department of Mechanics and Mathematics
Tbilisi State University
2, University St., Tbilisi 0143
Georgia