Int. Journal of Math. Analysis, Vol. 7, 2013, no. 51, 2535 - 2548
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2013.38205
On Uniform Convergence of Double Sine Series
under Condition of p-Supremum Bounded
Variation Double Sequences
Moch Aruman Imron
Department of Math, Faculty of Math and Sciences
University of Brawijaya, Malang and
Graduate School, Faculty of Mathematics and Sciences
University of Gadjah Mada, Yogyakarta, Indonesia
Ch. Rini Indrati and Widodo
Department of Math, Faculty of Math and Sciences
University of Gadjah Mada,Yogyakarta, Indonesia
Correspondence should be addressed to: Moch Aruman Imron, [email protected]
Copyright © 2013 Moch Aruman Imron et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
The classical theorem on the uniform convergence of sine series with
monotone decreasing coefficients have been proved by Chaundy and Jollife in
1916. Recently, the monotone decreasing coefficiets has been generalized by
classes of Mean Bounded variation Sequences, Supremum Bounded Variation
Sequence and p-Supremum Bounded Variation Sequences. In two variables,
class of Mean Bounded Variation Double Sequences and Supremum Bounded
Variation Double Sequences were studied under the uniform convergence of
double sine series. We shall generalize those results by defining class of
p-Supremum Bounded Variation Double Sequences and study uniform
convergence of double sine series.
2536
Moch Aruman Imron, Ch. Rini Indrati and Widodo
Mathematics Subject Classification: 42A20, 42A32, 42B99
Keywords: double sine series, p-supremum bounded variation double series,
uniform convergence
1. Introduction
The classical theorem on the uniform convergence of sine series with
monotone decreasing coefficients have been proved by Chaundy and Jollife in
1916 [1, 12] as stated in Theorem 1.1.
Theorem 1.1. Suppose that ⊂ 0,∞
is decreasing tending to zero. The
necessary and sufficient conditions for the uniform convergence of the series
is lim→ 0.
∑
sin 1.1
Many classes of sequences have been introduced by many reseachers such as
Tikhonov [11], Zhou [10], Korus [7]. These classes are more general than the
classes of monotone decreasing coeffecients (1.1). The definition of the classes
of SBVS (Supremum Bounded Variation Sequences), SBVS (Supremum Bounded
Variation Sequences), SBVS2 (Supremum Bounded Variation Sequences of second
type) are the following.
,-.
,2
$
| |, / ' &7
|∆ | +
sup )
⊆ !|∃$ % 0, ∃& ' 1 ∶ )
/ 23-/56 2 2 ,-.
,2
$
|∆ | +
| |, / ' 17
⊆ !|∃$ % 0, ∃9 ↗ ∞ ∶ )
sup )
/ 23;-
2 where ∆ < = and C, >, & depend only , [x] the greatest
integer that is less than or equal x.
Imron et. al. [4] introduced classes of ?@A?B (p-Supremum Bounded
Variation Sequences) and ?@A?2B (p-Supremum Bounded Variation Sequences
of second type) stated in Definition1.2:
Definition 1.2. Let - and C C- be two sequences of complex
and positive numbers, respectively. A couple , C
is said to be
(i) p-Supremum Bounded Variation Sequences, written , C
∈ ?@A?B , if
there exist a positive constant C and & ' 1 such that
On uniform convergence of double sine series
2537
B /B
∑,-.
+ Fsup23GHJ ∑,2
2 C K ,/ ' 1,
- | < = | E
-
I
(ii) p- Supremum Bounded Variation Sequences of second type, written,
, C
∈ ?@A?2B , if there exist a positive constant C and 9
⊂ 0, ∞
tending monotonically to infinity depending only on , such that
L
,2
B M
∑,-.
- | < = | + - Nsup23;-
∑2 C O,/ ' 1,
for 1 + P Q ∞,
As the single sine series (1.1) we have double sine series of two variables.
Let RS T ⊆ ! and consider the double sine series of the form
E
∑2
S ∑ S UV/WUV/ 1.2
As an idea in one variable, to investigate the uniform convergence of double sine
series, the coefficients of the series (1.2) are supposed to be member of class of
general monotone double sequences. In two variables, Supremum Bounded
Variation Double Sequences of first type (SBVSDS1) and Supremum Bounded
Variation Double Sequences of second type
(SBVSDS2)
have been
introduced by Korus [8].
A double sequence RS T ⊆ ! belongs
if there exists C > 0 and integer > ' 2 and 9 X
, 9, X
, 9Y X
, each one
Definition 1.3.
to class SBVSDS1,
converges to infinity, all of them depend only RS T such that
∑,2.
S2 ZΔ\ S- Z +
E
2
max;L_
`a`b;L2
∑,a
SaZS- Z ,c ' >,/ ' 1,
,e
∑,-.
- |Δ\ 2 | + - max;d -
`e`b;d -
∑e|2 | ,c ' 1,/ ' >,
E
,-.
∑,2.
S2 ∑- ZΔ S Z +
where
E
2-
,e
supa=e3;f 2=-
∑,a
Sa ∑eZS Z , c, / ' >,
∆\ S S < S=, ,
∆\ S S < S,= ,
∆ S ∆\ N∆\ S O ∆\ N∆\ S O S < S=, < S,= g S=,= .
Definition 1.4. A double sequence RS T ⊆ ! belongs to class SBVSDS2,
if there exists C > 0 and integer > ' 1 and 9X
, converging to infinity,
depending only RS T such that
Moch Aruman Imron, Ch. Rini Indrati and Widodo
2538
∑,2.
S2 ZΔ\ S- Z +
E
2
supa3;2
∑,a
SaZS- Z ,c ' >,/ ' 1,
,e
∑,-.
- |Δ\ 2 | + supe3;-
∑e|2 | ,c ' 1,/ ' >,
,-.
∑,2.
S2 ∑- ZΔ S Z +
E
-
E
2-
,e
supa=e3;2=-
∑,a
Sa ∑eZS Z , c, / ' >.
Related to double sine series of (1.2), Korus and Moricz [9] also consider the
regular convergence of double sequence stated in Definition 1.5.
Definition 1.5. A double sequence RS T ⊆ ! is
the sums
regularly convergence
if
∑2
S ∑ S
converge to a finite number as m and n tend to infinity independently of each
other, moreover
∑
S S- , / 1,2,3, … , and ∑ 2 , c 1, 2, 3, …
are convergent.
In the present paper, we construct classes of p-Supremum Bounded Variation
Double Sequences of first type ?@Ak?1B and p-Supremum Bounded Variation
Double Sequences of seconnd type ?@Ak?2B and investigate some of those
two classes. Futhermore, we investigate the uniform regular convergence of
the sine double series.
2. Uniform Convergence of Double Sine Series
In this section, we construct class of p-Supremum Bounded Variation
Double Sequences First Type and Second Type. Furthemore we investigate the
relations between those and also study other properties of class of p-Supremum
Bounded Variation Double Sequences. The Goal of this paper is to study the
uniform regular corvergence of double sine series in Theorem 2.14.
Definition 2.1. Let RS T and C RCS T be two double sequences of
complex and positive numbers, respectively. A couple , C
is said to be class of
p-Supremum
Bounded Variation Double Sequences first type
written
, C
∈ ?@Ak?1B
, if there exists C > 0 and integer > ' 2 and
9 X
, 9, X
, 9Y X
, each one converges to infinity,all of them depend only
RS T such that
B /B
(i) l∑,2.
S2 ZΔ\ S- Z m
+ 2 max;L_
`a`b;L 2
∑,a
Sa CS- ,c ' >,/ ' 1,
E
On uniform convergence of double sine series
/B
B
(ii) N∑,-.
- |Δ\ 2 | O
+ - max;d -
`e`b;d -
∑,e
e C2 ,c ' 1,/ ' >,
E
B /B
,-.
(iii) l∑,2.
S2 ∑- ZΔ S Z m
for 1 + P Q ∞.
2539
+
E
,e
supa=e3;f 2=-
∑,a
Sa ∑e CS
2-
, c, / ' >,
Definition 2.2. Let RS T and C RCS T be two double sequences of
complex and positive numbers, respectively. A couple , C
is said to be class of
p-Supremum Bounded Variation Double Sequences second type written
, C
∈ ?@Ak?2B , if there exists C > 0 and integer > ' 1 and 9X
,
converging to infinity, depending only RS T such that
B /B
(i) l∑,2.
S2 ZΔ\ S- Z m
/B
B
(ii) N∑,-.
- |Δ\ 2 | O
+ 2 supa3;2
∑,a
Sa CS- ,c ' >,/ ' 1,
E
+ - supe3;-
∑,e
e C2 ,c ' 1,/ ' >,
E
B /B
,-.
(iii) l∑,2.
S2 ∑- ZΔ S Z m
for 1 + P Q ∞.
Theorem 2.3.
+
E
,e
supa=e3;2=-
∑,a
Sa ∑e CS
2-
If 1 + P Q ∞, then
, c, / ' >,
?@Ak?1B ⊊ ?@Ak?2B .
Proof:
First, we prove that ?@Ak?1B ⊆ ?@Ak?2B , let , C
∈ ?@Ak?1B and put
9/
inf 9 /
, 9, /
, 9Y /
, 9 / g 1
, 9, / g 1
, 9Y / g 1
, … ,
for
1 + P Q ∞. From Definition 2.1 and Definition 2.2 , C
∈ ?@Ak?2B , so we
have ?@Ak?1B ⊆ ?@Ak?2B .
The second, we prove that ?@Ak?1B ⊋ ?@Ak?2B . From idea of example [8]
dr
can be constructed in the following way. Set cq 2N, O for s 1, 2, 3, … and
0
w
u 1
S 0
v
u1
t0
if1 + W Q c ,
ifW cq ,
if/S Q W Q cq, ,
ifcq, + W Q 2cq, ,
if2cq, + W Q cq=. We define the sequence RCS T, where CS S , for every j. By Theorem 2.1. [6],
NRS T, RCS TO ∈ ?@Ak?B with constant $ 4 and 9W
G,J.
S
Moch Aruman Imron, Ch. Rini Indrati and Widodo
2540
The second example
0 if1 + W Q c ,
cq., ifW cq ,
S, 0 ifcq Q W Q cq ,
.Y/,
v
c, + W + 2cS,
u cq= if S
t 0 if2cq, Q W Q cq=. w
u
We define the sequence RCS, T, where CS, S, , for every j. By Theorem 2.4. [6],
NRS, T, RCS, TO ∈ ?@Ak?2B with constant $ 2 and 9W
W/, .
we define NRS T, RCS TO ∈ ?@Ak?2B . Thus
S 0 for W ' 1, ' 3.
It is clear that for n=2, by Definition 1.3 NRS- T, RCS- TO ∉ ?@Ak?1B . Futhermore
the second and third condition of Definition 1.4 are satiesfied for $ 4 and
9X
X/, , this means NRS- T, RCS- TO ∈ ?@Ak?2B . Thus
?@Ak?1B ⊊ ?@Ak?2B . ∎
Theorem 2.4. If 1 + P Q { Q ∞, then ?@Ak?1B ⊆ ?@Ak?1| .
Finally,
Proof: Given , C
∈ ?@Ak?1B and for 1 + P Q { Q ∞,
(i) From proof Theorem 3.1 [4] by replacing ∆S to ∆S- obtained
}
BM
∑,2.
S2 ZΔ\ S- Z
}
B M
N∑,2.
S2 ZΔ\ S- Z O ,
+
Hence by Definition 2.1 we have
,2.
|
/|
~ ) ZΔ\ S- Z 
S2
,2.
B
,a
$
+ ~ ) ZΔ\ S- Z  +
max
) CS- .
c ;L_
`a`b;L 2
S2
B
Sa
(ii) Similarly by proof (i) and replacing ∆S- to ∆2 and Definition 2.1
we have
,-.
/|
€ ) |Δ\ 2 || 
-
,-.
B
,e
$
+ € ) |Δ\ 2 |B  +
max
) C2 .
/ ;LH
`e`b;L -
-
e
(iii) Similarly by proof (i) and replacing ∆S- to ∆S with double
sumation and
Definition 2.1 we have
,2. ,-.
|
/|
~ ) ) ZΔ S Z 
S2 -
,2. ,-.
B
/B
+ ~ ) ) ZΔ S Z 
S2 -
On uniform convergence of double sine series
2541
,a ,e
From (i),( ii) and (iii),
$
+
sup
) ) CS c/ a=e3;f2=-
Sa e
, C
∈ ?@Ak?1| , then ?@Ak?1B ⊆ ?@Ak?1| . ∎
Theorem 2.5. If 1 + P Q { Q ∞, then ?@Ak?2B ⊆ ?@Ak?2| .
Proof: Similarly as proof of Theorem 2.4 and Definition 2.2.
Definition 2.6.
Let C RCS T be double sequences of positive numbers.
A class of p-Supremum Bounded Variation Double Sequences first type
of C, written ?@Ak?1B C , is defined as
R: , C
∈ ?@Ak?1B T.
(ii) A class of p-Supremum Bounded Variation Double Sequences second type
of C,
written ?@A?2B C , is defined as
R: , C
∈ ?@Ak?2B T.
(i)
By Theorem 2.3 , Theorem 2.4 and Theorem 2.5 and straight from Definition
2.6 we have Corollary 2.7, Corollary 2.8 and Corollary 2.9.
Corollary 2.7.
Corollary 2.8.
Corollary 2.9.
If 1 + P Q ∞, then ?@Ak?1B C
⊆ ?@Ak?2B C
.
If 1 + P Q ∞, then ?@Ak?1B C ⊆ ?@Ak?2B C .
If 1 + P Q ∞, then ?@Ak?1B C ⊆ ?@Ak?2B C .
Some properties of ?@A?k2B , 1 + P Q ∞, are stated below. Theorem 2.10 give
a sufficient condition for a couple of sequences in ?@Ak?2ƒ to be have bounded
variation introduced by Hardy [3]. The notion of bounded variation of double
sequence of real or complex term is defined as follows:
The double sequence - is bounded variation in /, if
(i). - is for every fixed value of n and k, bounded variation in n and k.
(ii). the summation ∑
- ∑|∆- | is convergent.
Theorem 2.10.
(i)
Let , C ∈ ?@Ak?2B , 1 + P Q ∞.
for c ∈ „, c
L
,.
M
sup ∑,a
Sa CS-
If
a3;2
decreasing monotone.
(ii)
for / ∈ „, /
,.
L
M
sup ∑,e
e C2 ,
e3;-
decreasing monotone.
(iii)
for c, /
∈ „…„, c/
decreasing monotone.
,.
L
M
,e
supa=e3;2=-
∑,a
Sa ∑e CS ,
Moch Aruman Imron, Ch. Rini Indrati and Widodo
2542
then
(i)
(ii)
,./B
sup ∑,a
c ∑
S2Z∆\ S- Z + $ †c
Sa CS- ‡,
/ ∑
-|∆\ 2 | + $ †/
L
M
,.
a3;2
sup ∑,e
e C2 ‡,
e3;-
(iii) c/ ∑
S2 ∑-Z∆ S Z + $ c/
holds respectively for p, 1 + P Q ∞.
,.
L
M
,e
Nsupa=e3;2=-
∑,a
Sa ∑e CS O,
Proof. We prove this Theorem in three parts
(i)
Let , C
∈ ?@A?k2B and c,./B sup ∑,a
Sa CSa3;2
decreasing monotone for c ∈ „ . We denote
cˆ2 cNc./B O supa3;2
∑,a
Sa CS-
for every c ∈ „. Therefore
,
2.
c ∑
S2Z∆\ S- Z c ∑‹\ ∑S,‰ 2 Z∆\ S- Z
‰ŠL
L
M
2.
,
+ c ∑
Z∆\ S- Z m 2‹ ‹\ Œl∑S,‰
‰ŠL
‹
+ c ∑
‹\2 c
+ $ Ž ∑
‹\
.
d‰ _ ,‰ 2
L
M
B
L
.M
 2.1
L
B M
,
2.
l∑S,
‰ 2 Z∆\ S- Z m ‰ŠL
,‰
2_
+ $ ∑‹\ ,‰ 2.2
Ž
+ $Nc,./B O supa3;2
∑,a
Sa CS- , $ 2$ .
Ž
By Holder inequality obtained (2.1) and (2.2) obtained because of the decreasing
monotone condition of Theorem 2.10 (i).
(ii)
(iii)
The proof of (ii) is similar with the proof of (i) by replacing m to n and
∆\ S- to ∆\ 2 .
Let , C ∈ ?@A?k2B and
L
,e
c/
M Nsupa=e3;2=-
∑,a
Sa ∑e CS O decreasing monotone for
c, /
∈ „„. We denote
,.
c/2- c/ Fc/
.
L
M
,e
supa=e3;2=-
∑,a
Sa ∑e CS K
for every c, / ∈ „, then we write
-.
,
2. ,
c/ ∑
S2 ∑-Z∆ S Z c/ ∑‹\ ∑’\ ∑S,‰ 2 ∑,‘ - Z∆ S Z
‰ŠL
‘ŠL
On uniform convergence of double sine series
2543
L
M
,
-.
,
2.
’
+ c/ ∑
‹\ ∑’\ ∑S,‰ 2 l∑,‘ - Z∆ S- Z m 2 / +
L
L
.M
2.3
L
B
.
,‰ŠL 2. ,‘ŠL -.
‹=’
∑,‘ - Z∆ S Z mM
c/ ∑
c/
M l∑S,
‰2
‹\ ∑’\ 2
+ $ Ž ∑
‹\ ∑’\
+
B
‘ŠL
‰ŠL
N,‰Š‘ 2-O“d‰Š‘ _H
,‰Š‘
2-
“
$ Ž ∑‹\ ∑’\ ,‰Š‘_H 2.4
+ $ Fc/
,.
L
M
,e
Ž
supa=e3;2=-
∑,a
Sa ∑e CS K , $ 4$ .
By Holder inequality obtained (2.3) and (2.4) obtained because of the decreasing
monotone condition of Theorem 2.10 (iii).
The proof of this Theorem is complete.
Theorem 2.11.
(i)
(ii)
(iii)
Let , C ∈ ?@Ak?2B , 1 + P Q ∞. If
the summation c
the summation /
,.
,.
L
M
L
M
sup ∑,a
Sa CS- , tending to 0, for / → ∞,
a3;2
sup ∑,e
e C2 , tending to 0, for / → ∞,
e3;-
L
,.
the summation c/
M
,e
supa=e3;2=-
∑,a
Sa ∑e CS ,
tending to 0, for c g / → ∞.
then, the sequence RS T is bounded variation.
Proof.
(i)
Given , C
∈ ?@Ak?2B and let
cˆ2- c †c
tending to 0 for c → ∞.
L
M
.
sup ∑,a
Sa CS- ‡
a3;2
We denote
”2 sup•32 –ˆ•- , for fixed n,
then
”2 → 0, for c → ∞ .
For every — % 0 ,
there exists
c\ ∈ „ such that |”2 | Q — for c ' c\ and from the proof of
Theorem 2.10 (i), we obtained
Ž c ∑
S2Z∆\ S- Z + $ ∑‹\
Thus
∑
S2™ Z∆\ S- Z + $
Therefore RS- TS
(ii)
d‰ _H ,‰ 2
,‰
˜H™
-™
+ $ Ž ∑
‹\
Q $
š
-™
˜_
,‰
+ $”2 , $ 2$ Ž .
is bounded variation.
Proof of (ii) similar with the proof of (i) by replacing m to n and ∆\ Sto ∆\ 2 . Therefore 2 is bounded variation.
Moch Aruman Imron, Ch. Rini Indrati and Widodo
2544
(iii)
Given , C ∈ ?@Ak?2B and let
c/2- c/ ~c/
.
B
sup
a=e3;2=-
,a ,e
) ) CS 
Sa e
tending to 0 for c g / → ∞. We denote
”2- sup U›‹’ ,
‹32,’3-
then ”2- → 0, for c g / → ∞. For every — % 0, there exists
c\ , /\ ∈ „…„ such that |”2- | Q — for c, /
' c\ , /\ and
from the proof of Theorem 2.10 (iii). we obtined
Ž c/ ∑
S2 ∑-Z∆ S Z + $ ∑‹\ ∑’\
Then we have
Ž ∑
S2 ∑-Z∆ S Z + $ ∑‹\ ∑’\
+ $ ) )
Ž
Thus
‹\ ’\
,‰Š‘
N,‰Š‘ 2-O“d‰Š‘ _H
,‰Š‘ 2-
”2‹=’
2 c/
∑
S2 ∑-Z∆ S Z + $
N,‰Š‘ 2-O“d‰Š‘_H
+$
Z˜_™ H™ Z
2™ -™
”2,
c/
+$2
.
$ 4$ Ž .
š
™ -™
for c, /
' c\ , /\ .
Therefore from proof (i), (ii) and (iii), the double sequence RS TS,
is
bounded variation. The proof of this Theorem is complete.
Lemma 2.12. Let RS T ⊆ 0, ∞
and , C
∈ ?@A?k2B , 1 + P Q ∞.
condition (i), (ii) and (iii) of Theorem 2.11. are satiesfied, then
WS → 0UW g → ∞.
If
Proof. From idea proof of Lemma 3 [7], for t be anbitrary natural numbers and
U c g 1, c g 2, … , 2c, we have
‹.
2’ ∑‹.
2 ∆\ ’ g ‹’ + ∑2|∆\ ’ | g ‹’ .2.5
Next, an analogous argument, for s be anbitrary natural numbers and
› / g 1, / g 2, … , 2/, we have
’.
‹- ∑’.
2 ∆\ ‹ g ‹’ + ∑2|∆\ ‹ | g ‹’ .2.6
Finally, let m, n be natural numbers. A double version of the above argument, for
U c g 1, c g 2, … , 2c and › / g 1, / g 2, … , 2/, we have
’.
2- ∑‹.
2 ∑- ∆  g 2’ g ‹- < ‹’
On uniform convergence of double sine series
2545
’.
+ ∑‹.
2 ∑-|∆  | g 2• g ‹- .2.7
If we add up all inequalities in (2.7) and from (2.5), (2.6), then by proof of
Theorem 2.11, we have
‹. ’.
‹.
’.
2 -
2
2
c/2- + c/ ) )|∆  | g c ) ∆\ ‹ g / ) ∆\ ‹ + 3$—
for c, /
' c\ , /\ . Thus WS → 0UW g → ∞.
Lemma 2.13. If , C ∈ ?@Ak?2B , 1 + P Q ∞ and condition (i) , (ii) and (iii)
of Theorem 2.11. are satisfied, then
(i)
(ii)
(iii)
c/ ∑
S2 ∑-Z∆ S Z → 0,
c/ ∑
S2 Uup3- Z∆\ S- Z → 0, c/ ∑
- U P3- |∆\ 2 | → 0. Proof.
(i)
From proof of Theorem 2.11. we have
c/ ∑
S2 ∑-Z∆ S Z + $”2- .
for any c, /
∈ „…„. By condition (iii) of Theorem 2.11, therefore
c/ ∑
S2 ∑-Z∆ S Z → 0, as c g / → ∞.
From Lemma 2.12, we obtained
W|S | → 0UW g → ∞.
(ii)
Further
∆\ 2- ∑
- ∆ 2 ,∆\ 2- ∑S2 ∆ S-, 2.8
(iii)
for c, / 1, 2,3, … . By (2.8) we have
sup3- Z∆\ S Z U P3- Z∑
‹ ∆ S‹ Z + ∑-Z∆ S Z
W, / 1, 2, 3, ….Thus we have
∑
S2 sup3- Z∆\ S Z + ∑S2 ∑-Z∆ S Z
and an application of (i) yields
c/ ∑
S2 U P3- Z∆\ S Z → 0.
A similar argument by replacing ∆\ to ∆\ , obtained (iii).
Theorem 2.14. If , C
∈ ?@Ak?2B , 1 + P Q ∞ and condition (i) , (ii) and
(iii) of Theorem 2.11 are satisfied, then the series (1.2) converges uniformly,
regularly at , ¢
for all 0 + , ¢ + £.
Proof.
To prove the uniformly regularly convergent of double sine series (1.2),
Moch Aruman Imron, Ch. Rini Indrati and Widodo
2546
first by letting the single sine series with coefficients of double sequence RS T
∑
S S- sin W, / 1, 2, 3, …,2.9
∑
2 sin ¢, c 1, 2, 3, ….2.10
Furthermore, by the condition Definition 2.2 (i) and (ii) it can be proved that for
any c ' 1, 2 , C ∈ ?@A?2B and for any / ' 1, lRS- T- , Cm ∈
?@A?2B . Hence by condition (i) and (ii) and Theorem 3.1. shown by Imron et. al
[4] implies the uniform convergence of the series (2.9) and (2.10). The second,
we show that the double series (1.2) is regularly convergence at (x,y) for all
0 + , ¢ + £.
(i)
Following an idea from Moricz [2], for , ¢
¥ 0, 0 we represent the
rectangular partial sums
2- , ¢
∑2
S ∑ S UV/WUV/ ,c, / ' 1,
of series (1.2) to perform a double summation by part yields
2- , ¢
2
-
) ) ¦S∗ ¦∗ ¢
∆ S
S -
2
g ) ¦S∗ ¦-∗ ¢
∆\ S,-= S
∗ ∗ ∗
g ) ¦2
¦-∗ ¢
2.11
¦ ¢ ∆\ 2=, g 2=,-= ¦2
where ¦-∗ conjugate Dirichlet kernel defined as
¦-∗ ∑- sin and |¦-∗ | + ¬­
for ∈ 0, £6.
¨©‹Ldª.¨©‹N-=LdOª
,‹«- Ldª
,/ ' 1
Given condition (iii), by Lemma 2.13 we obtained
thus
∑
S ∑Z∆ S Z Q ∞
∗
∗
∑
S ∑Z¦S ¦ ¢
∆ S Z Q ∞2.12
Given condition (i), (ii) and (iii), by Lemmma 2.12
ZS Z → 0, for W g → ∞2.13
and from (2.8) we have
∆\ S,-= ∑
-= ∆ S .
Further
∑
SZ∆\ S,-= Z + ∑S ∑-=Z∆ S Z → 0as/ → ∞.
This implies that, for all 0 Q , ¢ + £
∗
∗
∑2
S ¦S ¦- ¢ ∆\ S,-= → 0as/ → ∞, 2.14
On uniform convergence of double sine series
2547
uniformly in m. Similarly, for all 0 Q , ¢ + £
∗ ∗ ∑- ¦2
¦ ¢ ∆\ 2=, → 0asc → ∞, 2.15
uniformly in n. By (2.13)
∗ ∗ 2=,-= ¦2
¦- ¢ → 0asc g / → ∞.2.16
Consequently by (2.11), (2.12), (2.14), (2.15) and (2.16)
(ii)
2- , ¢
→ 0, asc g / → ∞.
For , ¢
0, 0
, ∑
S ∑ S UV/WUV/ 0.
From the first and second part of this proof and by Definition 1.5 it can be
concluded that series (1.2) is regular convergence. Futhermore by Lemma 2.12,
Lemma 2.13 and Theorem 1 [6], the regular convergence is uniformly at (x,y)
for all 0 + , ¢ + £. The proof is complete. ∎
The uniform regular convergence of the series in (1.2) in class of ?@Ak?1B for p, 1 + P Q ∞, is stated in Corollary 2.15. We abandon the proof, since it is
similar to the proof of Theorem 2.14.
Corollary 2.15.
(i)
(ii)
Let , C ∈ ?@Ak?1B , 1 + P Q ∞. If
the summation Fc
for c → ∞,
the summation F/
for / → ∞,
,.
,.
L
M
L
M
max;L2
`a`b;L2
∑,a
Sa CS- K ®1
max;d -
`e`b;d -
∑,e
e C2 K ®1
L
,e
the summation c/
M Nsupa=e3;f2=-
∑,a
Sa ∑e CS O ®1
for c g / → ∞,
then the series (1.2) is uniformly regularly convergence at , ¢
for all
0 + , ¢ + £.
(iii)
3. Conclusions
,.
In this paper we have introduced the classes of ?@Ak?1B and ?@Ak?2B .
We have investigated that
(i)
The class of ?@Ak?1B and ?@Ak?2B is more general than class
SBDS1 and class SBVDS2 respectively introduced by Korus [8] straight
from Definition 2.1 and Definition 2.2.
(ii)
Uniformly regularly convergence of (1.2) in Theorem 2.14 is more general
than Theorem 1 by Korus [8] and Theorem 1 by Korus and Moricz [9].
2548
Moch Aruman Imron, Ch. Rini Indrati and Widodo
Ackowledgments.
The authors gratefully acknowledge the support of the Department of
Mathematics, Faculty of Mathematics and Sciences University of Brawijaya and
Graduate School Department of Mathematics, Faculty of Mathematics and
Sciences, University of Gadjah Mada.
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[1]
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Received: August 25, 2013