Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 104592, 15 pages
doi:10.1155/2012/104592
Research Article
On Double Summability of Double Conjugate
Fourier Series
H. K. Nigam and Kusum Sharma
Department of Mathematics, Faculty of Engineering and Technology, Mody Institute of Technology
and Science, Deemed University, Laxmangarh 332311, Sikar, Rajasthan, India
Correspondence should be addressed to Kusum Sharma, kusum31sharma@rediffmail.com
Received 20 January 2012; Accepted 1 March 2012
Academic Editor: Ram U. Verma
Copyright q 2012 H. K. Nigam and K. Sharma. 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.
For the first time, a theorem on double matrix summability of double conjugate Fourier series is
established.
1. Introduction
The Fourier series of fx is given by
fx ∼
∞
a0 an cos nx bn sin nx.
2 n1
1.1
Conjugate to the series 1.1 is given by
∞
an cos nx − bn sin nx
1.2
n1
and is known as conjugate Fourier series. It is well known that the corresponding conjugate
function of 1.2 is defined as
fx 1
π
π
0
fx t − fx − t
dt.
2 tan1/2t
1.3
Let fx, y is integrable L over the square Q−π, −π; π, π and is periodic with
period 2π in each variable.
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The double Fourier series of a function fx, y which is analogue for two variables of
the series 1.1, is given by
∞ ∞
f x, y λm,n αm,n cos mx cos ny βm,n sin mx cos ny γm,n cos mx sin ny
m0 n0
δm,n sin mx sin ny
1.4
∞ ∞
λm,n Am,n x, y ,
m0 n0
αm,n
1
2
π
f x, y cos mx cos nydxdy
1.5
Q
with three similar expressions for m, n 0, 1, 2, . . . and for βm,n , γm,n and δm,n where Q
represents the fundamental square −π, −π; π, π.
One can associate three conjugate series to the double Fourier series 1.4 in the
following way:
∞
∞ λm,n −βm,n cos mx cos nyαm,n sin mx cos ny − δm,n cos mx sin nyγm,n sin mx sin ny ,
m1 n0
1.6
∞ ∞
λm,n −γm,n cos mx cos ny − δm,n sin mx cos nyαm,n cos mx sin nyβm,n sin mx sin ny ,
m0 n1
∞
∞ 1.7
δm,n cos mx cos ny − γm,n sin mx cos ny − βm,n cos mx sin ny αm,n sin mx sin ny ,
m1 n1
1.8
where λm,n 1, λm,0 λ0,n 1/2, m, n ≥ 1, λ0,0 1/4.
The conjugate functions f
1.8 are defined as
1
x, y, f
2
x, y and f
3
x, y corresponding to 1.7, and
1 π f x s, y − f x − s, y
ds,
f
x, y −
π 0
2 tan1/2s
π f x, y t − f x, y − t
2 1
dt,
f
x, y −
π 0
2 tan1/2t
1 π π f x s, y t − f x − s, y t − f x s, y − t − f x − s, y − t
ds dt.
f x, y 2
2 tan1/2s 2 tan1/2t
π 0 0
1.9
1 We will consider the symmetric square partial sum of series 1.8.
International Journal of Mathematics and Mathematical Sciences
3
∞
Let T am,j and S bn,k be two infinite triangular matrices. Let ∞
m0
n0 pm,n
m n
be a double series with sm,n j0 k0 pj,k as its m, nth partial sums. The double matrix
mean tm,n is given by
tm,n n
m am,j bn,k sj,k ,
1.10
j0 k0
∞
The double series ∞
m0
n0 pm,n with the sequence of m, nth partial sums sm,n is
said to summable by double matrix summability method or summable T, S if tm,n tends to
a limit s as m → ∞ and n → ∞.
The regularity conditions of double matrix summability means are given by
n
m am,j bn,k −→ 1 as m −→ ∞, n −→ ∞,
j0 k0
lim
m,n
lim
m,n
n am,j bn,k 0
k0
m am,j bn,k 0
for each j 1, 2, 3, . . . ,
1.11
for each k 1, 2, 3, . . .
j0
We write
ψ x, y ψ x, y; s, t f x s, y t − f x − s, y t − f x s, y − t f x − s, y − t ,
x y
ψs, tds dt,
Ψ x, y 0
Ψ1 x, t x0
ψs, tds,
0y
Ψ2 s, y ψs, tdt,
0
1
1
τ
integral part of ,
t
t
1
1
σ
integral part of ,
s
s
m
cos j 1/2 s
1 ,
K m s am,j
2π j0
sins/2
K n t n
cosk 1/2t
1 ,
bn,k
2π k0
sint/2
1.12
Important particular cases of the double matrix summability method are
i C, 1, 1 summability mean 1 if am,j 1/m 1for all m and bn,k 1/n 1 for all n;
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ii H, 1, 1 summability mean 5 if am,j 1/m − j 1 log m and bn,k 1/n − k 1 log n;
iii N, pm , qn summability mean 2 if am,j pm−j /Pm and bn,k qn−k /Qn , provided
n
Pm m
/ 0 and Qn k0 qk / 0.
j0 pj Double matrix summability method T, S is assumed to be regular throughout this
paper.
2. Main Theorem
Rajagopal 1 previously proved a theorem on the Nörlund summability of Fourier series.
Result of Rajagopal 1 contained various results due to Hardy 2, Hirokowa 3, Hirokowa
and Kayashima 4, Pati 5, Siddiqui 6, and Singh 7. Thereafter Sharma 8 proved
a theorem dealing with the harmonic summability of double Fourier series. The result of
Sharma 8 is a generalization of the theorem due to Hille and Tamarkin 9 for double Fourier
series and also is analogous to the theorem of Chow 10 for summability C, 1, 1 of the
double Fourier series. The theorem of Sharma 8 was generalized by Mishra 11 for double
Nörlund summability. The result Mishra 11 was generalized by Okuyama and Miyamoto
12. But nothing seems to have been done so far to study double matrix summability of
conjugate Fourier series. Therefore, the purpose of this paper is to establish the following
theorem.
and bn,k nk0 be two real nonnegative and nondecreasing sequences with
Theorem 2.1. Let am,j m
j0
j ≤ m and k ≤ n, respectively.
Let T am,j and S bn,k be two infinite triangular matrices with
am,j ≥ 0,
am,j 0,
j > m;
bn,k ≥ 0, bn,k 0, k > n;
σ
τ
am,j ;
Bn,τ bn,k ;
Am,σ j0
Am,m 1
2.1
k0
∀m ≥ 0;
Bn,n 1
∀n ≥ 0.
If the conditions
Ψ x, y x y
0
0
ψs, t ds dt
y
x
O
· ,
ξ1/x χ 1/y
π
x
,
ψ1 x, tdt O
ξ1/x
0
π
y
ψ2 s, y ds O
χ 1/y
0
2.2
2.3
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5
hold, then the double conjugate Fourier series 1.8 is double matrix T, S summable to, fx, y,
where
1
f x, y − 2
4π
π
0
1
ds dt
tan1/2s tan1/2t
2.4
at every point where these integrals exist provided ξx and χy are two positive monotonic
increasing functions of x and y such that ξm → ∞, as m → ∞ and χn → ∞, as n → ∞,
m
Am,s
ds O1,
sξs
1 n
Bn,t
dt O1,
tχt
1
m −→ ∞,
2.5
n −→ ∞.
3. Lemmas
Lemma 3.1. One has
1
K m s O
s
for 0 ≤ s ≤
1
.
m
3.1
Proof. For 0 ≤ s ≤ 1/m, sint/2 ≥ t/π and | cos mt| ≤ 1,
m
1 cos j 1/2 s K m s ≤ am,j
2 j0
sins/2
⎧
⎫
m
cos j 1/2 s ⎬
1 ⎨
≤
am,j
⎭
2 ⎩ j0
|sins/2|
≤
m
π
s
3.2
am,j
j0
1
.
O
s
Lemma 3.2. One has
K n t O
1
t
for 0 ≤ t ≤
1
.
n
3.3
Proof. This can be proved similar to Lemma 3.1.
Lemma 3.3 see 2. If am,u is non-negative and non-decreasing with μ, then, for 0 ≤ a ≤ b ≤
∞, 0 ≤ s ≤ π and any m,
b
am,m−μ eim−μs OAm,σ .
μa
3.4
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International Journal of Mathematics and Mathematical Sciences
Lemma 3.4. If bn,v is non-negative and non-decreasing with ν, then, for 0 ≤ a ≤ b ≤ ∞, 0 ≤ t ≤ π
and any n,
b
in−νt OBn,τ .
bn,n−ν e
νa
3.5
Proof. This is similar to Lemma 3.3.
Lemma 3.5. One has
K m s O
Am,σ
s
for 0 <
1
< s ≤ π.
m
3.6
Proof. Since 1/m < s ≤ π, sins/2 ≥ s/π,
m
cos
j
s
1/2
1
am,j
K m s ≤
2π j1
sins/2
m
1 im−j1/2s ≤ am,m−j Re e
2s j1
m
1 ≤ am,m−j Re eim−js eis/2 2s j1
m
1 im−js Re am,m−j e
2s j1
1
O
· OAm,σ by Lemma 3.3
s
Am,σ
O
.
s
3.7
Lemma 3.6. One has
K n t O
Bn,τ
,
t
for 0 <
1
< π.
n
3.8
Proof. It can be proved similar to Lemma 3.5 but using Lemma 3.4.
4. Proof of The Theorem
The j, kth partial sums sj,k x, y of the series 1.8 is given by
sj,k
1
x, y − f x, y 4π 2
π π
0
0
cos j 1/2 s cosk 1/2t
ds dt.
ψs, t
sins/2
sint/2
4.1
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7
Then,
m n
am,j bn,k sj,k x, y − f x, y
j0 k0
1
4π 2
m n
cos j 1/2 s cosk 1/2t
·
ds dt
φs, t
am,j bn,k
sins/2
sint/2
j0 k0
π π
0
0
4.2
or
π
tm,n x, y − f x, y ψs, tK m sK n tds dt
0ν
η ν π π η π π
· ψs, tK m sK n tds dt
0
0
0
η
I1 I 2 I 3 I 4
ν
0
ν
4.3
η
say .
We consider
1/m 1/n ν
I1 0
0
ν
1/n 1/m η
1/m
0
I1.1 I1.2 I1.3 I1.4
0
1/n
1/m
η · ψs, tK m sK n tds dt
1/n
4.4
say ,
where s ≤ ν, t ≤ η and 2.2 holds.
Now consider
I 1.1 1/m 1/n
ψs, tK m sK n tds dt
0
1/m
1/n 1
1
O
ψs, tds dt by Lemma 3.1 and 3.2
s
t
0
0
1/m 1/n
Omn
ψs, tds dt
0
0
0
1 1
,
Omnψ
m n
1
by 2.2
Omno
mnξmχn
o1,
as m −→ ∞, n −→ ∞.
4.5
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Now,
|I1.2 | ≤
1/n
ν
1/m
1/n
Oψs, tK m sK n tds dt
0
ν
ψs, t Am,1/s ds by Lemma 3.5
s
0
1/m
1/n ν
Am,1/s
1
ψs, t
ds
by Lemma 3.2
dt
t
s
0
1/m
1/n
1/n Am,1/ν
1
On
ψ1 ν, tdt OmnAm,m , t dt
ψ1
ν
m
0
0
1/n ν
d Am,1/s ψs, tds
dt
On
s
0
1/m ds
I1.2.1 I1.2.2 I1.2.3
say .
≤
|Kn t|dt
4.6
Then,
I1.2.1 I1.2.2 On
1/n
ψ1 ν, tdt Omn
0
1/n
ψ1
0
1
, t dt
m
1
1 1
OnΨ ν,
,
OmnΨ
n
m n
1/n
1/mn
ν
·
Omno
Ono
ξ1/ν χn
ξmχn
o1,
4.7
as m −→ ∞, n −→ ∞.
Next,
1/n
I1.2.3
ν
1/n ν
Am,1/s
Ψ1 s, t d On
dt
Ψ1 s, t
ds On
dt
Am,1/s ds
2
s
ds
s
0
1/m
0
1/m
ν 1/n
Am,1/s
Ψ1 s, tdt
ds
On
s2
1/m
0
ν 1/n
1 d
Ψ1 s, t
Am,1/s ds
On
s ds
1/m
0
ν
ν
1 Am,1/s
1 1 d
Ψ1 s,
ds On
Ψ1 s,
On
Am,1/s ds
2
n
n
s
ds
s
1/m
1/m
ν
Am,1/s
1
s
·
o
ds
On
ξ1/s nχn
s2
1/m
ν
1
1 d
s
·
o
On
Am,1/s ds
ξ1/s nχn s ds
1/m
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m
m
Am,s
1
1
1 d
ds o
o
Am,s ds
χn
χn
1/ν sξs
1/ν ξs ds
1
1
o1 o
o
Am,n χn
ξmχn
o1,
9
as m −→ ∞, n −→ ∞.
4.8
Thus,
I1.2 o1,
as m −→ ∞, n −→ ∞.
4.9
I1.3 o1,
as m −→ ∞, n −→ ∞.
4.10
Similarly
Now,
ν
η
ψs, t Am,1/s · Bn,1/t dtds
s
t
1/m 1/n
mBn,1/η Am,m Ψ 1/m, η
Bn,1/η Am,1/ν Ψ ν, η
O
−
νη
η
ν
Bn,1/η
d Am,1/s
nBn,n Am,1/ν Ψν, 1/n
ds −
−
ψ s, η
η
ds
s
ν
1/m
ν
1 d Am,1/s
1 1
,
nBn,n
ds
mn Bn,n Am,m Ψ
Ψ s,
m n
n ds
s
1/m
ξ
Am,1/ν ν
d Bn,1/t
d Bn,1/t
1
dt − mAm,m
,t
dt
Ψ1 s, t
Ψ2
ν
dt
t
m
dt
t
1/m
1/n
ν η
d Am,1/s d Bn,1/t
−
Ψs, t
dt ds
ds
s
dt
t
1/m 1/n
I1.4.1 I1.4.2 I1.4.3 I1.4.4 I1.4.5 I1.4.6 I1.4.7 I1.4.8 I1.4.9
say .
4.11
|I1.4 | O
Then,
mBn,1/η Am,m Ψ 1/m, η
Bn,1/η Am,1/ν Ψ ν, η
O
νη
η
Bn,1/η Am,1/ν
η
1
· o
o mBn,1/η
mξm χ 1/η
α1/νβ 1/η
Bn,1/η Am,1/ν
η
1
o
· o Bn,1/η
ξm χ 1/η
ξ1/νχ 1/η
I1.4.1 I1.4.2
10
I1.4.3
International Journal of Mathematics and Mathematical Sciences
1
o1 o1
ξm
o1, as m −→ ∞, n −→ ∞ by the regularity of T, S,
Bn,1/η ν
d Am,1/s
ds
−
Ψ s, η
η
ds
s
1/m
ν Bn,1/η
Am,1/s
η
s
o
· ds
η
ξ1/s χ 1/η
s2
1/m
ν
Bn,1/η
η
1 d
s
o
· Am,1/s ds
o
η
ξ1/s χ 1/η
s ds
1/m
ν
ν
Bn,1/η
Bn,1/η
Am,1/s
d
1
o
ds o
Am,1/s ds
χ 1/η
χ 1/η
1/m sξ1/s
1/m ξ1/s ds
m
m
Bn,s/η
Bn,1/η
Am,s
1
ds o
dAm,s o
χ 1/η
χ 1/η
1/ν sξs
1/ν ξs
Bn,1/η
Bn,1/η
o
O1 o
OAm,m χ 1/η
χ 1/η
o1,
as m −→ ∞, n −→ ∞.
4.12
Thus, we get
I1.4.3 o1,
as m −→ ∞, n −→ ∞.
4.13
I1.4.4 o1,
as m −→ ∞, n −→ ∞.
4.14
Similarly
Now,
I1.4.5
1 1
,
mnBn,n Am,m Ψ
m n
1
o mnBn,n Am,m
mnξmχn
o1,
as m −→ ∞, n −→ ∞.
4.15
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11
Consider
ν
1 d Am,1/s
ds
Ψ s,
n ds
s
1/m
ν
ν
1 Am,1/s
1 1 d
nBn,n
Ψ s,
ds nBn,n
Ψ s,
Am,1/s ds
2
n
n
s
ds
s
1/m
1/m
ν
A
1
s
m,1/s
·
o
ds
nBn,n
ξ1/s
s2
nχ
η
1/m
ν 1 d
s
nBn,n
Am,1/s ds
s ds
ξ1/snχ η
1/m
ν
ν
Am,1/s
nBn,n
1
d
1
ds o
o
Am,1/s ds
nχn
χn
1/m sξ1/s
1/m ξ1/s ds
I1.4.6 nBn,n
o1,
4.16
as m −→ ∞, n −→ ∞.
As similar to I1.4.3 ,
I1.4.7 o1,
as m −→ ∞, n −→ ∞.
4.17
I1.4.8 o1,
as m −→ ∞, n −→ ∞.
4.18
As similar to I1.4.6 ,
Now,
ν
η
Am,1/s
Bn,1/t 1 d 1 d
I1.4.9 O
Ψs, t
Am,1/s ·
Bn,1/t dt ds
s ds
t dt
s2
t2
1/m 1/n
I1.4.9.1 I1.4.9.2 I1.4.9.3 I1.4.9.4
say .
4.19
Then,
I1.4.9.1 I1.4.9.2
ν
η
Am,1/s Bn,1/t
O
Ψs, t
·
dt ds
s2
t2
1/m 1/n
ν η
Am,1/s 1 d O
·
Ψs, t
dt
ds
B
n,1/t
t dt
s2
1/m 1/n
ν η Am,1/s Bn,1/t
t
s
O
·
o
·
dt
ds
ξ1/s χ1/t
s2
t2
1/m 1/n
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International Journal of Mathematics and Mathematical Sciences
ν η Am,1/s 1 d s
t
·
·
o
O
dt
ds
B
n,1/t
ξ1/s χ1/t
t dt
s2
1/m 1/n
ν η
Am,1/s Bn,1/t
dt ds
o
sξ1/st
χ1/t
1/m 1/n
ν η
Am,1/s
d
o
Bn,1/t dt ds
1/m 1/n sξ1/sχ1/t dt
m
n
Am,s
Bn,t
ds
dt
o
1/ν sξs
1/η tχt
m
n
Am,s
1 d
ds
o
Bn,1/t dt
1/ν sξs
1/η χt dt
n
d
O1
o1 o1
Bn,1/t dt
dt
1/η
o1,
as m −→ ∞, n −→ ∞.
4.20
Similarly,
I1.4.9.3 o1,
as m −→ ∞, n −→ ∞.
4.21
Now,
ν
η
1 d 1 d
Am,1/s
Bn,1/t dt ds
s ds
t dt
1/m 1/n
ν η 1 d t
1 d
s
·
o
Am,1/s ·
Bn,1/t dt ds
t dt
1/m 1/n ξ1/s χ1/t s ds
ν η d
d
1
o
Am,1/s
Bn,1/t dt ds
dt
1/m 1/n ξ1/s · χ1/t ds
η
ν
d
d
1
1
o
Am,1/s ds
Bn,1/t dt
1/m ξ1/s ds
1/n χ1/t dt
η
ν
d
d
O1
O1
o
Am,1/s ds
Bn,1/t d
ds
dt
1/m
1/n
I1.4.9.4 O
o1,
Ψs, t
4.22
as m −→ ∞, n −→ ∞.
Hence,
I1.4 o1,
as m −→ ∞, n −→ ∞.
4.23
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13
Therefore,
I1 o1,
as m −→ ∞, n −→ ∞.
4.24
Now, 1/m < ν < π, 1/n < η < π, thus we obtain
|I3 | ≤
π
|Km s|ds
ν
I3.1 I3.2
1/n
ψs, t|Kn t|dt 0
π
ν
say .
|Km s|ds
ν
ψs, t|Kn t|dt
1/n
4.25
Using Lemma 3.2 and Lemma 3.5, we have
1/n
Am,σ
ψs, t 1 dt
ds
s
t
ν
0
π
1/n
Am,σ
ψs, tdt
On
ds
s
0
νπ 1
On
ds
Ψ2 s,
n
0 1
O
χn
I3.1 π
o1,
as m −→ ∞, n −→ ∞.
Using Lemma 3.5 and Lemma 3.6,
η
Bn,τ
Am,σ
ψs, t
ds
dt
O
s
t
ν
1/n
η
π Bn,τ η
d Bn,τ
O
dt
ds Ψ2 s, t
−
Ψs, t
t
dt
t
1/n
ν
1/n
π Bn,1/n
1
O
− Ψ2 s,
ds Ψ2 s, η
nBn,n
η
n
ν
π
I3.2
4.26
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International Journal of Mathematics and Mathematical Sciences
η
π
d Bn,τ
ds
Ψ2 s, t
dt
dt
t
ν
1/n
π π
Bn,1/n
1
ds On
O
Ψ2 s, η
Ψ2 s,
ds
η
n
ν
0
η
π
d Bn,τ
dt
ds
Ψ2 s, t
O
dt
t
ν
1/n
o1 o1 o1similar to I1.4.9 o1,
as m → ∞, n → ∞.
4.27
Hence,
I3 o1,
as m −→ ∞, n −→ ∞.
4.28
I2 o1,
as m −→ ∞, n −→ ∞.
4.29
Similarly,
By the regularity conditions of matrix method and Riemann-Lebesgue theorem,
I4 o1,
as m −→ ∞, n −→ ∞.
4.30
Therefore,
tm,n x, y − f x, y o1
4.31
This completes the proof of the theorem.
References
1 C. T. Rajagopal, “On the Nörlund summability of Fourier series,” Mathematical Proceedings of the
Cambridge Philosophical Society, vol. 59, pp. 47–53, 1963.
2 G. H. Hardy, “On the summability of Fourier series,” Proceedings London Mathematical Society, vol. 12,
pp. 365–372, 1913.
3 H. Hirokawa, “On the Nörlund summability of Fourier series and its conjugate series,” Proceedings of
the Japan Academy, vol. 44, pp. 449–451, 1968.
4 H. Hirokawa and I. Kayashima, “On a sequence of Fourier coefficients,” Proceedings of the Japan
Academy, vol. 50, pp. 57–62, 1974.
5 T. Pati, “A generalization of a theorem of Iyengar on the harmonic summability of Fourier series,”
Indian Journal of Mathematics, vol. 3, pp. 85–90, 1961.
6 J. A. Siddiqui, “On the harmonic summability of Fourier series,” Proceedings of the National Institute of
Sciences of India, vol. 28, pp. 527–531, 1948.
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National Institute of Sciences of India A, vol. 29, pp. 65–73, 1963.
International Journal of Mathematics and Mathematical Sciences
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8 P. L. Sharma, “On the harmonic summability of double Fourier series,” Proceedings of the American
Mathematical Society, vol. 9, pp. 979–986, 1958.
9 E. Hille and J. D. Tamarkin, “On the summability of Fourier series. I,” Transactions of the American
Mathematical Society, vol. 34, no. 4, pp. 757–783, 1932.
10 Y. S. Chow, “On the Cesàro summability of double Fourier series,” The Tohoku Mathematical Journal,
vol. 5, pp. 277–283, 1954.
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Institute of Mathematics. Academia Sinica, vol. 13, no. 3, pp. 289–295, 1985.
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