Bulletin of TICMI
Vol. 16, No. 2, 2012, 1–20
Uniform Convergence of Double Fourier-Legendre series of
Functions of Bounded Generalized Variation
Ushangi Goginava
I.Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University,
University str. 2, Tbilisi 0128, Georgia
e-mail: zazagoginava@gmail.com
(Received February 28, 2012; Accepted August 21, 2012)
The Uniform convergence of double Fourier-Legendre series of function of bounded Harmonic
variation and bounded partial Λ-variation are investigated.
Keywords: Fourier-Legendre series, Generalized bounded variation, Uniform convergence
AMS Subject Classification: 42C10.
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 ets., see [5, 18, 23, 24]). In
two dimensional case the class BV of functions of bounded variation was introduced
by Hardy [13].
Let f be a real function of two variables. Given intervals ∆ = (a, b), J = (c, d)
and points x, y from I := [−1, 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 an
arbitrary way and let Ω be the set of all such collections E. Denote by Ωn the set
of all collections of n nonoverlapping intervals ∆k ⊂ I.
2
2
For the sequence of positive numbers Λ = {λn }∞
n=1 and I := [−1, 1] we denote
ΛV1 (f ; I 2 ) = sup sup
y
ISSN: 1512-0082 print
c 2012 Tbilisi University Press
°
E∈Ω
X |f (∆i , y)|
i
λi
(E = {∆i }) ,
2
Bulletin of TICMI
ΛV2 (f ; I 2 ) = sup sup
X |f (x, Jj )|
λj
x F ∈Ω
j
ΛV1,2 (f ; I 2 ) = sup
F, E∈Ω
(F = {Jj }),
X X |f (∆i , Jj )|
i
j
λi λj
.
Definition 1.1: We say that the function f has bounded Λ-variation on I 2 and
write f ∈ ΛBV , if
ΛV (f ; I 2 ) := ΛV1 (f ; I 2 ) + ΛV2 (f ; I 2 ) + ΛV1,2 (f ; I 2 ) < ∞.
We say that the function f has bounded partial Λ-variation and write f ∈ P ΛBV
if
P ΛV (f ; I 2 ) := ΛV1 (f ; I 2 ) + ΛV2 (f ; I 2 ) < ∞.
¡ 1 2¢
Definition 1.2: We say
that
the
function
f
is
continuous
in
Λ , Λ -variation
¡
¢ ¡ ¢
on I 2 and write f ∈ C Λ1 , Λ2 V I 2 , if
¡
¢
¡
¢
lim Λ1n V1 f ; I 2 = lim Λ2n V2 f ; I 2 = 0
n→∞
n→∞
and
¡
¢
¡
¢
¡
¢
¡
¢
lim Λ1n , Λ2 V1,2 f ; I 2 = lim Λ1 , Λ2n V1,2 f ; I 2 = 0,
n→∞
n→∞
ª∞
©
© ª∞
where Λin := λik k=n = λik+n k=0 , i = 1, 2.
If λn ≡ 1 (or if 0 < c < λn < C < ∞, n = 1, 2, . . .) the classes ΛBV and P ΛBV
coincide with the Hardy class BV and PBV respectively. 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 λn = ∞.
n→∞
(1)
In the case when λn = n, n = 1, 2 . . . we say Harmonic Variation instead of
Λ-variation and write H instead of Λ (HBV , P HBV , HV (f ), ets).
The notion of Λ-variation was introduced by Waterman [24] in the one dimensional case, by Sahakian [21] in the two dimensional case. The notion of
bounded partial variation (P BV ) was introduced by Goginava [11] and the notion of bounded partial Λ-variation, by Goginava and Sahakian [12].
The statements of the following theorem is known.
Theorem D [Dragoshanski [9]] HBV = CHV .
Definition 1.3: Let Φ-be a strictly increasing continuous function on [0, +∞)
with Φ (0) = 0. We say that the function f has bounded partial Φ-variation on I 2
Vol. 16, No. 2, 2012
3
and write f ∈ P BVΦ , if
n
X
(1)
VΦ (f ) := sup sup
y
(2)
VΦ (f ) := sup
{Ii }∈Ωn i=1
m
X
sup
x {Jj }∈Ωm
j=1
Φ (|f (Ii , y) |) < ∞,
n = 1, 2, ...,
Φ (|f (x, Jj ) |) < ∞,
m = 1, 2, ....
In the case when Φ (u) = up , p ≥ 1, the notion of bounded partial p-variation
(class P BVp ) was introduced in [10].
In [12] it is proved that the following theorem is true.
Theorem GS Let Λ = {λn } and λn /n ≥ λn+1 / (n + 1) > 0, n = 1, 2, .... .
1) If
∞
X
λn
n=1
n2
< ∞,
then P ΛBV ⊂ HBV .
2) If, in addition, for some δ > 0
λn
λ [1+δ]
= O( n[1+δ] )
n
n
as
n→∞
and
∞
X
λn
n=1
n2
= ∞,
then P ΛBV 6⊂ HBV .
Corollary 1.4: P BV ⊂ HBV and P HBV 6⊂ HBV .
Definition 1.5 see [11]: The partial modulus of variation of a function f are the
functions v1 (n, f ) and v2 (m, f ) , defined by
v1 (n, f ) := sup sup
y
v2 (m, f ) := sup
n
X
{Ii }∈Ωn i=1
sup
m
X
x {Jk }∈Ωm
i=1
|f (Ii , y)| ,
|f (x, Jk )| ,
n = 1, 2, . . . ,
m = 1, 2, . . . .
For functions of one variable the concept of the modulus variation was introduced
by Chanturia [5].
4
Bulletin of TICMI
The following theorem is proved by Goginava and Sahakian [12].
Theorem GS2 If f ∈ B is bounded on I 2 and
p
vj (n, f )
< ∞,
3/2
n
n=1
∞
X
j = 1, 2,
then f ∈ HBV.
¡ ¢
C I 2 is the space of continuous functions on I 2 with the uniform norm
kf kC := max |f (x, y)| .
(x,y)∈I 2
¡ ¢
The partial moduli of continuity of a function f ∈ C I 2 in the C-norm are
defined by
ω1 (f ; δ) := max {|f (x, y) − f (s, y)| , x, y, s ∈ I, |x − s| ≤ δ} ,
ω2 (f ; δ) := max {|f (x, y) − f (x, t)| , x, y, t ∈ I, |y − t| ≤ δ} ,
while the mixed modulus of continuity is defined as follows:
ω1,2 (f ; δ1 , δ2 ) : = max {|f (x, y) − f (s, y) − f (x, t) + f (s, t)| ,
x, y, s, t ∈ I, |x − s| ≤ δ1 , |y − t| ≤ δ2 } .
2.
Fourier-Legendre Series
Let pn (x) be the Legendre orthonormal polynomial of degree n. If f is an integrable
function on I 2 , then Fourier-Legendre series of f is the series
∞ X
∞
X
fb(n, m) pn (x) pn (y) ,
n=0 m=0
where
Z1 Z1
fb(n, m) :=
f (s, t) pn (s) pm (t) dsdt
−1 −1
is the (n, m)th Fourier coefficient of the function f .
The rectangular partial sums of double Fourier-Legendre series are defined by
SM N f (x, y) :=
N
−1 M
−1
X
X
n=0 m=0
fb(n, m) pn (x) pm (y) .
Vol. 16, No. 2, 2012
5
It is easy to show that
Z1 Z1
f (s, t) Kn (x, s) Km (y, t) dsdt,
SM N f (x, y) =
(2)
−1 −1
where
Kn (x, s) :=
n−1
X
pk (s) pk (x) .
k=0
The Chrestoffel-Darboux formula is well-know (see ([22]))
Kn (x, t) =
γn−1 pn−1 (t) pn (x) − pn−1 (x) pn (t)
.
γn
x−t
(3)
Since
γn−1
≤1
γn
and
|pn (x)| ≤
c
(1 − x2 )1/4
, x ∈ (−1, 1)
(4)
from (3) we have
|Kn (x, t)| ≤
c
.
(5)
(−1 ≤ s < x < 1) ,
(6)
(−1 < x < s ≤ 1) ,
(7)
(−1 < x < 1) ,
(8)
|x − t| (1 − x2 )1/4 (1 − t2 )1/4
In [4, 19] it is proved that the following estimations hold
¯ s
¯
¯Z
¯
¯
¯
c
¯ Kn (x, t) dt¯ ≤
¯
¯
1/4
¯
¯ n (x − s) (1 − x2 )
−1
¯
¯ 1
¯
¯Z
¯
¯
c
¯ Kn (x, t) dt¯ ≤
¯
¯
1/4
¯ n (s − x) (1 − x2 )
¯
s
Zx
|Kn (x, t)| dt ≤
x−
1+x
n
c (1 + x)
(1 − x2 )1/2
6
Bulletin of TICMI
x+ 1−x
n
Z
|Kn (x, t)| dt ≤
x
3.
c (1 − x)
(1 − x2 )1/2
(−1 < x < 1) .
(9)
Convergence of double Fourier-Legendre series
The well known Dirichlet-Jordan theorem (see [25]) states that the trigonometric
Fourier series of a function f (x), x ∈ [0, 2π) of bounded variation converges at
every point x to the value [f (x + 0) + f (x − 0)] /2.
Hardy [13] generalized the Dirichlet-Jordan theorem to the double trigonometric
Fourier series. He proved that if the function f (x, y) has bounded variation in
the sense of Hardy (f ∈ BV ), then the double trigonometric Fourier series of
the continuous function f converges uniformly on [0, 2π]2 . The author [11] has
proved that in Hardy’s theorem there is no need to require the boundedness of
V1,2 (f ); moreover, it is proved that if f is a continuous function and has bounded
partial variation (f ∈ P BV ) then its double trigonometric Fourier series converges
uniformly on [0, 2π]2 .
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 [21], Dyachenko [6–8], Bakhvalov [1], Sablin [20].
For the one-dimensional Fourier-Legendre series the convergence of partial sums
of functions Harmonic bounded variation and other bounded generalized variation
were studied by Agakhanov, Natanson [2], Bojanic [4], Belenkiı̆ [3], Kvernadze
[15–17], Powierska [19].
In this paper we prove that the following are true.
¡ ¢ T ¡ 2¢
C I . Then
Theorem 3.1 : Let ε > 0 and f be a function of CHV I 2
the double Fourier-Legendre series of the function f uniformly converges to f on
[−1 + ε, 1 − ε]2 .
Theorem D and Theorem 3.1 imply
¡ ¢ T ¡ 2¢
C I . Then
Theorem 3.2 : Let ε > 0 and f be a function of HBV I 2
the double Fourier-Legendre series of the function f uniformly converges to f on
[−1 + ε, 1 − ε]2 .
Theorem GS1 and Theorem 3.2 imply
T ¡ ¢
Theorem 3.3 : Let f ∈ P ΛBV (I 2 ) C I 2 with
∞
X
λj
j=1
j2
< ∞,
λj
↓ 0.
j
Then the double Fourier-Legendre series of the function f uniformly converges to
f on [−1 + ε, 1 − ε]2 ,ε > 0.
n
o
¡ ¢ T ¡ 2¢
Corollary 3.4: If f ∈ P log1+δn(n+1) BV I 2
C I for some δ > 0. Then
the double Fourier-Legendre series of the function f uniformly converges to f on
[−1 + ε, 1 − ε]2 .
Theorem GS2 and Theorem 3.2 imply
Vol. 16, No. 2, 2012
7
¡ ¢
Theorem 3.5 : Let f ∈ C I 2 and
p
vj (n, f )
<∞
n3/2
n=1
∞
X
j = 1, 2.
Then the double Fourier-Legendre series of the function f uniformly converges to
f on [−1 + ε, 1 − ε]2 ,ε > 0.
¡ ¢
Corollary 3.6: Let f ∈ f ∈ C I 2 and v1 (k, f ) = O (k α ) , v2 (k, f ) =
¡ β¢
O k , 0 < α, β < 1. Then the double Fourier-Legendre series of the function
f uniformly converges to f on [−1 + ε, 1 − ε]2 ,ε > 0.
T ¡ ¢
Corollary 3.7: Let f ∈ P BVp C I 2 , p ≥ 1. Then the double Fourier-Legendre
series of the function f uniformly converges to f on [−1 + ε, 1 − ε]2 ,ε > 0.
4.
Proofs of Main Results
Proof : [Proof of Theorem 3.1] Denote
sj := x +
j (1 − x)
, j = 1, 2, ..., n, x ∈ (−1, 1) ,
n
(10)
ti := y −
i (1 + y)
, i = 1, 2, ..., m, y ∈ (−1, 1)
m
(11)
g (s, t) := f (s, t) − f (x, y) .
(12)
Then from (2) we can write
Smn f (x, y) − f (x, y)
(13)
Z1 Z1
g (s, t) Kn (x, s) Km (y, t) dsdt
=
−1 −1

Zx Zy
Z1 Zy
=
x −1
−1 −1
−1 y

 (g (s, t) Kn (x, s) Km (y, t) dsdt)
+
+
+
Z1 Z1
Zx Z1
x
y
= I1 + I2 + I3 + I4 ,
I1


Zs1 Zt1 Zs1 Zy Z1 Zt1 Z1 Zy
 (g (s, t) Kn (x, s) Km (y, t) dsdt)
=
+
+
+
x −1
x t1
s1 −1
= II1 + II2 + II3 + II4 .
s1 t1
(14)
8
Bulletin of TICMI
For II4 we have
Zy

(g (s, t) − g (sj , t)) Kn (x, s) ds Km (y, t) dt

II4 =

sZj+1
n−2
X
(15)
j=1 sj
t1
Zy


Z1

+
t1
Zy
(g (s, t) − g (sn−1 , t)) Kn (x, s) ds Km (y, t) dt
sn−1

n−1
X

+
Kn (x, s) ds Km (y, t) dt
g (sj , t)
j=1
t1

sZj+1
sj
= II41 + II42 + II43 .
From (4) and (5) we have
|II42 | ≤ 2 kf kC
Zy m−1
X
Z1
|pj (t) pj (y)| dt
j=0
t1
mdt
≤ c kf kC
y− 1+y
m
(1 −
Z1
sn−1
t2 )1/4 (1
− y 2 )1/4
ds
(s − x) (1 − x2 )1/4 (1 − s2 )1/4
Z1
≤ c (ε) kf kC
x+
≤
(16)
sn−1
Zy
×
|Kn (x, s)| ds
n−1
n
(1−x)
c (ε) kf kC
= o (1)
n3/4
ds
(1 − s)1/4
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
From (5), (10) and (11) we obtain
|II41 |

sj+1
Zy n−2
X Z
≤ 
t1
j=1 sj
Zy


≤ c (ε) m
t1

|g (s, t) − g (sj , t)|
(s − x) (1 − x2 )1/4 (1 − s2 )1/4
n−2
X
sZj+1
j=1 sj
ds |Km (y, t)| dt

|g (s, t) − g (sj , t)|
(sj − x) (1 − sj+1 )1/4
ds dt
(17)
Vol. 16, No. 2, 2012
Zy

1
j=1
j (n − j)1/4
t1
Zy
= c (ε) n5/4 m
t1
= c (ε) n5/4 m

1−x
Zn n−2
X |g (s + sj , t) − g (sj , t)| 

ds dt

j (n − j)1/4
j=1



+c (ε) n5/4 m



t1




1−x
Zn
X
|g (s + sj , t) − g (sj , t)|
j (n − j)
1≤j<n/2
0
Zy
t1
sj

t1
≤ c (ε) nm
|g (s, t) − g (sj , t)| ds dt
0
Zy
Zy

sZj+1
n−2
X

≤ c (ε) n5/4 m
9
1/4

1−x
Zn
X
|g (s + sj , t) − g (sj , t)|
1/4
n/2≤j<n−1
0
j (n − j)
+c (ε) n1/4 m
X
1≤j<n/2
Zy



t1

ds dt

1−x
Zn
0

ds dt
|g (s + sj , t) − g (sj , t)| 
ds dt
j

1−x
Zn
X
|g (s + sj , t) − g (sj , t)|
1/4
0
n/2≤j<n−1
(n − j)

ds dt.
It is easy to show that
X
1≤j<n/2
|g (s + sj , t) − g (sj , t)|
j
(18)


 X |g (s + s , t) − g (s , t)|
X
|g (s + sj , t) − g (sj , t)| 
j
j
+
≤ min

1≤k<n 
j
j
k≤j<n
1≤j<k
½ µ
¶
¾
¡
¢
1
≤ c (ε) min ω1 f ;
log (k + 1) + {j + k} V1 f ; I 2
1≤k<n
n
= o (1)
as n → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
On the other hand,
X
|g (s + sj , t) − g (sj , t)|
n/2≤j<n−1
(n − j)1/4
1
n3/4
≤
X
n/2≤j<n−1
|g (s + sj , t) − g (sj , t)|
n−j
(19)
10
Bulletin of TICMI
½ µ
¶
¾
¡
¢
1
2
log (k + 1) + {j + k} V1 f ; I
≤ c min ω1 f ;
1≤k<n
n
= o (1)
as n → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Combining (17)-(19), we obtain that
II41 = o (1)
as n → ∞
(20)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Applying the Abel transformation we obtain
II43


sk+1
Zy
n−1
X Z
= g (s1 , t)
Kn (x, s) ds Km (y, t) dt
(21)
k=1 sk
t1


sk+1
Zy n−2
n−1
X
X Z
(g (sj+1 , t) − g (sj , t))
+ 
Kn (x, s) ds Km (y, t) dt
j=1
t1
k=j+1 sk


Z1
Zy
= g (s1 , t) Kn (x, s) ds Km (y, t) dt
t1
Zy
s1

n−2
X

+
t1

Z1
(g (sj+1 , t) − g (sj , t))
j=1
Kn (x, s) ds Km (y, t) dt
sj+1
= II431 + II432 .
It is easy to show that
c (ε) m
|II431 | ≤
n (s1 − x)
Zy
|f (s1 , t) − f (x, y)| dt
t1
Zy
|f (s1 , t) − f (s1 , y)| dt
≤ c (ε) m
t1
Zy
+c (ε) m
|f (s1 , y) − f (x, y)| dt
t1
½ µ
¶
¶¾
µ
1
1
≤ c (ε) ω1 f,
+ ω2 f,
= o (1) as n, m → ∞
n
m
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
(22)
Vol. 16, No. 2, 2012
11
From (7), (18) and (19) we obtain
|II432 | ≤ c (ε)
Zy n−2
X
t1
j=1
≤ c (ε) sup
|g (sj+1 , t) − g (sj , t)|
|Km (y, t)| dt
(sj+1 − x) n
n−2
X
t∈[t1 ,y] j=1
(23)
|g (sj+1 , t) − g (sj , t)|
j
= o (1) as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
From (21)-(23) we have
II43 = o (1) as n, m → ∞
(24)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Combining (15), (20), (16) and (24) we conclude that
II4 = o (1) as n, m → ∞
(25)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Analogously, we can prove that
II1 = o (1) as n, m → ∞
(26)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
For II2 we can write
Zs1 Zy
|II2 | ≤
|f (s, t) − f (x, y)| |Kn (x, s)| |Km (y, t)| dsdt
(27)
x t1
½ µ
¶
¶¾
µ
1
1
≤ c (ε) ω1 f,
+ ω2 f,
= o (1) as n, m → ∞
n
m
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
We can write
II3
=
n−1
X m−1
X
(28)
sZj+1 Zti
j=1 i=1 sj t
i+1
g (s, t) Kn (x, s) Km (y, t) dsdt
12
Bulletin of TICMI
=
sZj+1 Zti
n−1
X m−1
X
(g (s, t) − g (sj , t) − g (s, ti ) + g (sj , ti ))
j=1 i=1 sj t
i+1
×Kn (x, s) Km (y, t) dsdt
+
sZj+1 Zti
n−1
X m−1
X
(g (sj , t) − g (sj , ti )) Kn (x, s) Km (y, t) dsdt
j=1 i=1 sj t
i+1
+
sZj+1 Zti
n−1
X m−1
X
(g (s, ti ) − g (sj , ti )) Kn (x, s) Km (y, t) dsdt
j=1 i=1 sj t
i+1
+
n−1
X m−1
X
sZj+1 Zti
g (sj , ti )
j=1 i=1
Kn (x, s) Km (y, t) dsdt
sj ti+1
= III1 + III2 + III3 + III4 .
For III3 we have
III3
=
(29)
n−2
X m−1
X
sZj+1 Zti
(g (s, ti ) − g (sj , ti )) Kn (x, s) Km (y, t) dsdt
j=1 i=1 sj t
i+1
+
m−1
X
Z1 Zti
(g (s, ti ) − g (sn−1 , ti )) Kn (x, s) Km (y, t) dsdt
i=1 sn−1 t
i+1
= III31 + III32 .
Applying the Abels transformation we get
III31
=
n−2
X m−1
X
(30)
sZj+1 Zti
(f (s, t1 ) − f (sj , t1 )) Kn (x, s) Km (y, t) dsdt
j=1 i=1 sj t
i+1
+
n−2
X m−2
X m−1
X
sZj+1 Ztk
(f (s, ti+1 ) − f (sj , ti+1 )
j=1 i=1 k=i+1 sj t
k+1
−f (s, ti ) + f (sj , ti )) Kn (x, s) Km (y, t) dsdt
=
n−2
X
sZj+1Zt1
j=1 sj −1
(f (s, t1 ) − f (sj , t1 )) Kn (x, s) Km (y, t) dsdt
Vol. 16, No. 2, 2012
+
n−2
X m−2
X
13
sZj+1 Z
ti+1
j=1 i=1 sj
(f (s, ti+1 ) − f (sj , ti+1 )
−1
−f (s, ti ) + f (sj , ti ) Kn (x, s)) Km (y, t) dsdt
= III311 + III312 .
From (5), (6), (18) and (19) we obtain
|III311 |
¯ t
¯
sZj+1
¯Z 1
¯ n−2
¯
¯X
|f (s, t1 ) − f (sj , t1 )|
ds
≤ c (ε) ¯¯ Km (y, t) dt¯¯
(sj − x) (1 − sj+1 )1/4 (1 + sj )1/4
¯
¯ j=1
(31)
sj
−1
n−2
c (ε) n5/4 X
≤
m (y − t1 )
sZj+1
|f (s, t1 ) − f (sj , t1 )|
j (n − j)1/4
j=1 sj
ds
1−x
= c (ε) n5/4
Zn n−2
X |f (s + sj , t1 ) − f (sj , t1 )|
0
= o (1)
j (n − j)1/4
j=1
ds
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 ,
|III312 |
¯t
¯ sj+1
i+1
¯ Z
n−2
X m−2
X ¯¯ Z
¯
¯
¯
≤
K
(y,
t)
dt
|f (s, ti+1 ) − f (sj , ti+1 )
m
¯
¯
¯
¯
j=1 i=1
sj
−1
−f (s, ti ) + f (sj , ti ) Kn (x, s)| ds
≤ c (ε)
n−2
X m−2
X
j=1 i=1
1
m (y − ti+1 )
−f (s, ti ) + f (sj , ti )|
≤ c (ε)
n−2
X m−2
X
j=1 i=1
sZj+1
|f (s, ti+1 ) − f (sj , ti+1 )
sj
ds
(sj − x) (1 − sj+1 )1/4
1
n5/4
i j (n − j)1/4
sZj+1
|f (s, ti+1 ) − f (sj , ti+1 )
sj
−f (s, ti ) + f (sj , ti )| ds
1−x
= c (ε)
Zn n−2
X m−2
X1
0
j=1 i=1
n5/4
i j (n − j)1/4
|f (s + sj , ti+1 ) − f (sj , ti+1 )
(32)
14
Bulletin of TICMI
−f (s + sj , ti ) + f (sj , ti )| ds
1−x
Zn
m−2
X
X
= c (ε)
1≤j<n/2 i=1
0
n5/4
ji (n − j)1/4
|f (s + sj , ti+1 ) − f (sj , ti+1 )
−f (s + sj , ti ) + f (sj , ti )| ds
1−x
Zn
m−2
X
X
+c (ε)
n/2≤j<n−1 i=1
0
n5/4
ji (n − j)1/4
|f (s + sj , ti+1 ) − f (sj , ti+1 )
−f (s + sj , ti ) + f (sj , ti )| ds
1−x
Zn
X
≤ c (ε) n
m−2
X
1≤j<n/2 i=1
0
1
|f (s + sj , ti+1 ) − f (sj , ti+1 )
ji
−f (s + sj , ti ) + f (sj , ti )| ds
1−x
Zn
X
+c (ε) n
0
m−2
X
n/2≤j<n−1 i=1
1
|f (s + sj , ti+1 ) − f (sj , ti+1 )
(n − j) i
−f (s + sj , ti ) + f (sj , ti )| ds
½
¶
µ
1 1
≤ min min ω1,2 f ; ,
log (k + 1) log (l + 1)
1≤k<n 1≤l<m
n m
¡
¢
¡
¢ª
+ {i + k} {j} V1,2 f ; I 2 + {i} {j + l} V1,2 f ; I 2
= o (1)
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Combining (30), (31) and (32) we have
III31 = o (1)
as n, m → ∞
(33)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Applying the Abel’s transformation we obtain
III32
=
m−1
X
Z1 Zti
(f (s, ti ) − f (sn−1 , ti )) Kn (x, s) Km (y, t) dsdt
i=1 sn−1 t
=
m−1
X
i+1
Z1 Zti
i=1 sn−1 t
+
(34)
(f (s, t1 ) − f (sn−1 , t1 )) Kn (x, s) Km (y, t) dsdt
i+1
m−2
X m−1
X
Z1 Ztk
i=1 k=i+1sn−1 t
k+1
(f (s, ti+1 ) − f (sn−1 , ti+1 )
Vol. 16, No. 2, 2012
15
−f (s, ti ) + f (sn−1 , ti )) Kn (x, s) Km (y, t) dsdt
Z1 tZm−1
(f (s, t1 ) − f (sn−1 , t1 )) Kn (x, s) Km (y, t) dsdt
=
sn−1 −1
+
m−2
X
ti+1
Z1 Z
(f (s, ti+1 ) − f (sn−1 , ti+1 )
i=1 sn−1 −1
−f (s, ti ) + f (sn−1 , ti )) Kn (x, s) Km (y, t) dsdt
= III321 + III322 .
Since
Z1
Z1
|Kn (x, s)| ds ≤ c (ε)
sn−1
sn−1
Z1
ds
≤ c (ε)
1/4
(1 − s)
sn−1
≤
ds
(s − x) (1 − s)1/4
c (ε)
n3/4
and
tZ
m−1
|Km (y, t)| dt ≤
−1
c (ε)
m3/4
for III321 we can write
|III321 | ≤
c (ε) kf kC
(nm)3/4
= o (1)
as n, m → ∞
(35)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
On the other hand,
|III322 | ≤
≤
c (ε)
sup
n3/4 s
m−2
X
c (ε)
sup
n3/4 s
m−2
X
i=1
i=1
¯t
¯
¯ Zi+1
¯
¯
¯
¯
|f (s, ti+1 ) − f (s, ti )| ¯
Km (y, t) dt¯¯
¯
¯
−1
|f (s, ti+1 ) − f (s, ti )|
m (ti+1 − x)
m−2
=
X |f (s, ti+1 ) − f (s, ti )|
c (ε)
sup
i
n3/4 s i=1
≤
¡
¢
c (ε)
HV2 f, I 2 = o(1) as n, m → ∞
3/4
n
(36)
16
Bulletin of TICMI
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
From (34), (35) and (36) we have
III32 = o (1)
as n, m → ∞
(37)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Combining (29), (33) and (37), we conclude that
III3 = o (1)
as n, m → ∞
(38)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Analogously we can prove that
III2 = o (1)
as n, m → ∞
(39)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
From (5) we have
|III1 |
≤ c (ε)
(40)
n−1
X m−1
X
sZj+1 Zti
|f (s, t) − f (sj , t) − f (s, ti ) + f (sj , ti )|
j=1 i=1 sj t
i+1
×
1
1
1
1
dsdt
s − x y − t (1 − s)1/4 (1 + t)1/4
1−x 1−y
≤ (nm)5/4
Zn Zm n−1
X m−1
X
0
0
j=1 i=1
1
1
1/4
j (n − j)
i (m − i)1/4
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )| dsdt.
We can write
n−1
X m−1
X
√
4
nm
j=1 i=1
≤
1
1
1/4
j (n − j)
i (m − i)1/4
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X 1
1≤j<n/2 1≤i<m/2
ji
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X
1
1
+ 3/4
m
j (m − i)1/4
1≤j<n/2 m/2≤i<m−1
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
(41)
Vol. 16, No. 2, 2012
+
1
n3/4
X
X
17
1
n/2≤j<n−1 1≤i<m/2
(n − j)1/4 i
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X
1
1
+
3/4
1/4
(nm) n/2≤j<n−1 m/2≤i<m−1 (n − j) (m − j)1/4
≤
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X 1
1≤j<n/2 1≤i<m/2
ji
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X
1
+
j (m − i)
1≤j<n/2 m/2≤i<m−1
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X
1
+
(n − j) i
n/2≤j<n−1 1≤i<m/2
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
X
X
1
+
(n − j) (m − j)
n/2≤j<n−1 m/2≤i<m−1
× |f (s + sj , t + ti ) − f (sj , t + ti ) − f (s + sj , ti ) + f (sj , ti )|
= IV1 + IV2 + IV3 + IV4 .
It is easy to show that
½
¶
µ
1 1
IV1 ≤ min min ω12 f ; ,
log (l + 1) log (r + 1)
1≤l<n 1≤r<m
n m
¡
¢
¡
¢ª
+ {i + l} {j} V1,2 f ; I 2 + {i} {j + r} V1,2 f ; I 2
= o (1)
(42)
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Analogously, we can prove that
IVi = o (1)
as n, m → ∞, i = 2, 3, 4
(43)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Combining (40), (41), (42) and (43) we get
III1 = o (1)
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
(44)
18
Bulletin of TICMI
Finally, we estimate III4 . By Abel’s transformation we have
III4
=
(45)
n−2
X m−2
X
(f (sj , ti ) − f (sj+1 , ti ) − f (sj , ti+1 ) + f (sj+1 , ti+1 ))
j=1 i=1
×
n−1
X
m−1
X
sZk+1 Ztl
Kn (x, s) Km (y, t) dsdt
k=j+1 l=i+1 sk tl+1
+
m−2
X
(f (s1 , ti+1 ) − f (s1 , ti ))
i=1
+
n−2
X
m−1
X
Zs1 Ztl
Kn (x, s) Km (y, t) dsdt
l=i+1 s0 tl+1
n−1
X
(f (sj+1 , t1 ) − f (sj , t1 ))
j=1
sZk+1Zt0
k=j+1 sk
sj+1 ti
n−1
X m−1
X Z Z
+g (s1 , t1 )
Kn (x, s) Km (y, t) dsdt
t1
Kn (x, s) Km (y, t) dsdt
j=1 i=1 sj t
i+1
= III41 + III42 + III43 + III44 .
From (6), (7), (18), (19) and (41) we have
|III41 |
≤
(46)
n−2
X m−2
X
|f (sj , ti ) − f (sj+1 , ti ) − f (sj , ti+1 ) + f (sj+1 , ti+1 )|
j=1 i=1
¯ 1 t
¯
¯ Z Zi
¯
¯
¯
× ¯¯
Kn (x, s) Km (y, t) dsdt¯¯
¯
¯
sj+1 −1
≤
n−2
X m−2
X
|f (sj , ti ) − f (sj+1 , ti ) − f (sj , ti+1 ) + f (sj+1 , ti+1 )|
j=1 i=1
×
≤
1
n (sj+1 − x) m (y − ti )
1
n−2
X m−2
X
j=1 i=1
= o (1)
|f (sj , ti ) − f (sj+1 , ti ) − f (sj , ti+1 ) + f (sj+1 , ti+1 )|
ij
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Vol. 16, No. 2, 2012
|III42 | ≤
m−2
X
|f (s1 , ti+1 ) − f (s1 , ti )|
19
(47)
i=1
x+ 1−x
n
Z
×
x
≤ c (ε)
¯t
¯
¯ Zi+1
¯
¯
¯
¯
|Kn (x, s)| ds ¯
Km (y, t) dt¯¯
¯
¯
−1
m−2
X
i=1
= o (1)
|f (s1 , ti+1 ) − f (s1 , ti )|
i
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Analogously, we can prove that
III43 = o (1)
as n, m → ∞
(48)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 ,
¯¯ t
¯
¯ 1
¯ ¯Z 1
¯
¯Z
¯¯
¯
¯
|III44 | ≤ |f (s1 , t1 ) − f (x, y)| ¯¯ Km (y, t) dt¯¯ ¯¯ Kn (x, s) ds¯¯
¯¯
¯
¯
s1
−1
µ µ
¶
¶¶
µ
1
1
≤ c (ε) ω1 f ;
+ ω2 f ;
n
m
= o (1)
(49)
as n, m → ∞
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
From (45)-(48) we have
III4 = o (1)
as n, m → ∞
(50)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
By (28), (38), (39), (44) and (50) we obtain
II3 = o (1)
as n, m → ∞
(51)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
From (14), (25), (26), (27) and (51) we conclude that
I1 = o (1)
as n, m → ∞
(52)
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Analogously we can prove that
Ii = o (1)
as n, m → ∞, i = 2, 3, 4
(53)
20
Bulletin of TICMI
uniformly with respect to (x, y) ∈ [−1 + ε, 1 − ε]2 .
Combining (13), (52) and (53) we complete the proof of Theorem 3.1.
¤
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