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www.ijma.info
A NOTE ON UNIFORM MATRIX SUMMABILITY
Shyam Lal
1, Mradul Veer Singh
2and Saurabh Porwal
3*1
Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi, (U.P.) - INDIA E-mail: [email protected]
2
Department of Mathematics, Lovely Professional University, (Punjab), INDIA E-mail: [email protected]
3
Department of Mathematics, UIET, CSJM University, Kanpur-208024, (U.P.), INDIA E-mail: [email protected]
(Received on: 07-12-11; Accepted on: 23-12-11)
________________________________________________________________________________________________ ABSTRACT
T
he purpose of the present paper is to establish a new result concerning uniform matrix summability of conjugate series of a Fourier series. Relevant connections of the results presented herewith various known results are briefly indicated.Key Words: Uniform triangular matrix summability, Conjugate series of Fourier series, Fourier coefficients, Nörlund summability.
AMS 2010 Mathematics Subject Classification: 42A24, 40C05.
________________________________________________________________________________________________
1. INTRODUCTION AND PRELIMINARIES:
Let
f
be2
π
periodic, Lebesgue integrable function with Fourier series given by
∞
=
+
+
≈
1
0
(
cos
sin
)
2
1
)
(
n
nx
n
b
nx
n
a
a
x
f
. (1.1)The conjugate series of series (1.1) is given by
1 1
( sin
n ncos )
n( )
n n
a
nt
b
nt
B t
∞ ∞
= =
−
=
−
. (1.2)Let
T
=
(
a
n,k)
be an infinite lower triangular matrix satisfying Silverman-Töeplitz [9] conditions of regularity i.e.(i)
→
→
∞
=
as n
a
n
k
n,k
1
0(ii)
a
n,k=
0
for k
>
n
and(iii)
=
≤
nk
n,k
M
a
0
where M is finite constant.
Let
∞
=0
)
(
n
n
x
u
be an infinite series defined in[
a
,
b
]
⊂
[
−
π
,
π
]
. Then
th partial sum of the series∞
=0
)
(
n
n
x
u
isgiven by
(
)
(
)
[
,
]
0b
a
x
x
u
x
S
n
n
=
∀
∈
=
ν
ν .
© 2012, IJMA. All Rights Reserved 234
If there exists a bounded function
S
(
x
)
such that
{
}
0
( )
n( )
( )
n n,k k
k
t x
a S x
S x
=
=
−
,
=
o
(1),
as n
→ ∞
,uniformly
∀
x
∈
[
a
,
b
]
then we say that the series∞ =0
)
(
n n
x
u
is summable (T) uniformly ina
≤
x
≤
b
to thesum
S
(
x
)
.Particular Cases:
Several authors such as ([1]-[4]), (see also [5]) studied the matrix summability method and obtained many interesting results.
The important particular cases of the triangular matrix means are:
(i) Cesàro mean of order 1 or (C, 1) mean if ,
1
1
n ka
k
n
=
∀
+
.(ii) Harmonic means when ,
1
=
(
1) log
n ka
n k
− +
n
.(iii) (C, ) means when ,
1
1
n kn
k
a
n
δ
δ
δ
δ
− + −
−
=
+
.(iv) (H, p) means when
1
, 1
0
1
log (
1)
(log)
(
1)
p q
n k p
q
a
k
n
− −
=
=
+
+
∏
.(v) Nörlund means when , n k n k
n
p
a
P
−
=
, where∞
=
=
0 kk
n
p
P
,P
n≠
0
.
(vi) Riesz means
(
N
,
p
n)
when , k n kn
p
a
P
=
,P
n≠
0
.(vii) Generalised Nörlund Means (N, p, q) when , n k k n k
n
p
q
a
R
−
=
. where∞
=
−
=
0 kk n k
n
p
q
R
,R
n≠
0
.We denote
S
n(
x
)
, the nth partial sum of the series (1.2).Let
1
1
( )
( )cot
2
2
n
n n
t
f
f x
t
dt
π
ψ
π
=
= −
, (1.3)
f
(
x
)
=∞ →
n
lim
f
n(
x
)
, (1.4)
ψ
(
t
)
=
f
(
x
+
t
)
−
f
(
x
−
t
)
, (1.5)
Ψ
=
tdu
u
t
0
)
(
)
(
ψ
, (1.6)
− = =
−
=
=
n
n k
k n o
k
k n n
n
a
a
A
τ τ
τ , ,
, , (1.7)
where
=
=
t
1
τ
integral part oft
1
and
=
+
=
n
k t
k n n
t
k
a
t
K
0 2
2 1 ,
sin
)
cos(
2
1
)
(
π
, (1.9)Saxena [6] discussed the uniform harmonic summability of conjugate series of a Fourier series in the following form:
Theorem: 1.1 If
Ψ
=
=
t t
t
o
du
u
t
10
log
)
(
)
(
ψ
, uniformly in a set E, ast
→
+
0
, then the series (1.2) issummable by harmonic means uniformly in E to the sum
f
(
x
)
, provided the limit (1.4) exists uniformly in E.Tripathi and Singh [8] extended the above result to the case of uniform Nörlund summability in the following form:
Theorem: 1.2 If the sequence
{ }
q
n is real, non-negative and monotonic non-increasing sequence of coefficients suchthat
Q
n→
∞
asn
→
∞
and the functionλ
(
t
)
,β
(
t
)
and)
(
)
(
t
t
t
β
λ
increase monotonically with
t
and)]
(
[
)
(
n
Q
nO
β
Q
nλ
=
asn
→
∞
, then if
Ψ
=
=
( )
)
(
)
(
)
(
1
0 τ
τ
β
λ
ψ
Q
q
o
du
u
t
tt
,
uniformly in a set E, as
t
→
+
0
, then the series (1.2) is summable(
N
,
q
n)
uniformly in E to the sumf
(
x
)
, at the pointt
=
x
, provided the limit (1.4) exists uniformly in E.2 MAIN THEOREM:
The purpose of this paper is to generalize the result of Saxena [6] and Tripathi and Singh [8] for uniform matrix summability method. In fact, we prove the following interesting result.
Theorem: 2.1 Let
T
=
(
a
n,k)
be an infinite triangular matrix such that the elements(
a
n,k)
are non-negative and non-decreasing withk
≤
n
and if
1 1 0
( )
( )
( )
=
log( )
tt
t
t
t
ψ
u
du o
β
Ψ
=
, (2.1)as
t
→
+
0
, uniformly in a setE
=
[ , ]
a b
, whereβ
(
t
)
is a positive function of t such that0
log
)
(
→
n
n
β
as
n
→ ∞
, then the conjugate series of a Fourier series (1.2) is summable (T) uniformly in E to0
1
( )
( )cot
2
2
t
f x
t
dt
π
ψ
π
= −
provided the limit (1.4) exist uniformly inE
=
[ , ]
a b
.To prove our main theorem we require the following lemmas.
3. LEMMAS:
Lemma: 3.1 If
(
a
n,k)
is a non-negative and non-decreasing withk
≤
n
, then)
(
)
cos(
,0 2
1
, nτ
n
k k
n
k
t
O
A
a
+
=
=
for
<
≤
t
<
δ
<
π
n
1
0
.Proof:
− = −
= =
+
+
+
≤
+
n
n k
k n n
k k n n
k k
n
k
t
a
k
t
a
k
t
a
τ τ
)
cos(
)
cos(
)
cos(
21,
0 2
1 ,
0 2
1
© 2012, IJMA. All Rights Reserved 236
− = =
− ≤ ≤ ≤
−
+
+
+
≤
n
n k
k n r
k n r k n
n
k
t
a
k
t
a
τ τ
τ
max
cos(
)
cos(
)
2
21, 0
2 1 0
, ,
(by Abel’s Lemma)
τ ,τ 2
2 ,
sin
)
1
sin(
2
nt t
n
n
A
r
a
+
+
≤
−
=
+
nk k
n
k
t
a
0 2
1
,
cos(
)
ττ
, ,
2
n n
n
A
t
a
+
≤
− . (3.1)Now
− = =
−
=
=
n
n k
k n k
k n n
n
a
a
A
τ τ
τ ,
0 , ,
=
a
n,n−τ+
a
n,n−τ+1+
...
+
a
n,n
≥
(
τ
+
1
)
a
n,n−τ
t
a
nn−τ≥
, ( sinceτ
=
[ ]
1t ).Therefore n,nτ
O
(
A
n,τ)
t
a
=
−
. (3.2)
By (3.1) and (3.2), we have
cos(
)
(
,)
0 2
1
, nτ
n
k k
n
k
t
O
A
a
+
=
=
.
Lemma: 3.2 If
(
a
n,k)
is non-negative and non-decreasing withk
≤
n
andK
n(
t
)
is given by (1.9) then=
t
A
O
t
K
n(
)
n,τ for<
≤
t
<
δ
<
π
n
1
0
.Proof: Since for
<
≤
t
<
δ
<
π
n
1
0
,π
t
t
≥
sin
,We have
=
+
=
n
k t
k n n
t
k
a
t
K
0 2
2 1 ,
sin
)
cos(
2
1
)
(
π
≤
+
=
t
k
a
n
k k n
t
cos(
)
sin
2
1
2 1
0 , 2
π
.
2
[
(
)
]
2
1
,τ
π
π
t
O
A
n≤
from Lemma 3.1
=
t
A
O
t
K
n(
)
n,τ .Hence the lemma is proved.
4. PROOF OF THE MAIN THEOREM:
It is well known that the integral formula for the kthpartial sum of conjugate series of a Fourier series (1.2) is given by, (see [7]):
S
x
t
k
t
tdt
tk
−
+
=
π
ψ
π
0 22 2
1
sin
cos
)
cos(
)
(
2
S
kx
f
x
t
k
tt
dt
+
=
−
π
ψ
π
0 22 1
sin
)
cos(
)
(
2
1
)
(
)
(
Now
a
{
S
x
f
x
}
t
a
k
tt
dt
n
k
k n k
n
k
k n
+
=
−
= =
π
ψ
π
0 22 1
0 , 0
,
sin
)
cos(
)
(
2
1
)
(
)
(
t
a
k
t
dt
t nk
k n
+
=
=
π
π
ψ
0 2
2 1
0 ,
sin
)
cos(
2
1
).
(
=
t
K
nt
dt
πψ
0
)
(
)
(
(4.1)So in order to prove our main theorem, we have to show that
(
)
(
)
(
1
)
0o
dt
t
K
t
n=
π
ψ
, uniformly in E. (4.2)We set
dt
t
K
t
dt
t
K
t
dt
t
K
t
dt
t
K
t
n n n nn n
+
+
=
π
δ δ
π
ψ
ψ
ψ
ψ
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1 1
0 0
=
I
1+
I
2+
I
3, say (4.3)Since limit (1.4) exists uniformly in E so
(
)
(
)
(
1
)
2
1
10
o
dt
t
K
t
nn
=
ψ
π
, uniformly in E. (4.4)Also, for
n
t
1
0
<
<
,=
−
+
nk t
t
k n
t
k
a
0 2
2 2
1
,
sin
cos
)
cos(
2
1
π
=+
=
n
k t
t t
k n
k
k
a
0 2
2 2 ,
sin
sin
)
1
sin(
2
2
1
π
=
+
≤
n
k t
t t
k n
k
k
a
0 2
2 2
,
sin
sin
.
sin
)
1
(
1
π
(Sincesin
kt
≤
k
sin
t
for
0
<
t
<
n1 )
=
+
=
n
k k
n
k
k
t
a
O
0
,
.
(
1
)
=
=
n
k k n
a
t
n
O
0 , 2
© 2012, IJMA. All Rights Reserved 238 Hence 1 1 0
( )
( )
n nI
=
ψ
t K
t dt
= =
+
−
+
=
n ndt
t
a
dt
t
k
a
t
t n k k n t t n k k n 1 1 0 2 0 , 0 2 2 2 1 0,
(
)
cot
2
1
sin
cos
)
cos(
2
1
)
(
ψ
π
π
ψ
I
t
O
n
t
dt
n)
(
.
)
(
2 0 1 1≤
ψ
+
o
(
1
)
, uniformly in E, (using (4.4) and (4.5))
O
n
t
dt
n≤
1 0)
(
).
(
ψ
+
o
(
1
)
, uniformly in E,
(
1
)
log
)
(
).
(
o
n
n
n
o
n
O
+
≤
β
, uniformly in E, (by condition (2.1))
(
1
)
log
)
(
o
n
n
o
+
≤
β
, uniformly in E,
=
o
(
1
)
, asn
→
∞
, uniformly in E (by hypothesis of theorem) (4.6)Now
I
t
K
t
dt
n n
=
δψ
1)
(
)
(
2t
dt
t
A
O
I
n n=
δ τψ
1)
(
.
).
1
(
,2 (by using Lemma 3.2)
=
Ψ
−
Ψ
t
dt
t
A
dt
d
t
t
A
O
n n n n δ τ δ τ 1 1)
(
.
)
(
.
).
1
(
, ,=
+
+
δ τ δ τ δτ
β
β
β
n n n n t t t t n t t n
A
d
t
t
dt
t
t
A
t
t
A
o
1 1 1)
(
)
log(
)
(
.
1
)
log(
)
(
.
)
log(
)
(
.
).
1
(
, 1 1 1 1 2 , 1 1 ,[ ]
[ ]
[ ]
+
+
+
=
n u n n u n n n nA
d
u
u
o
du
u
u
u
A
o
n
n
A
o
A
o
δ δδ
β
β
β
β
δ δ 1 1 1
)
(
log
)
(
log
)
(
log
)
(
log
, , , 1 1 ,( )
( )
[ ]
[ ]
[ ]
+
[ ]
+
+
+
=
+ = + = n k k n n k k n n na
n
n
o
a
o
n
n
A
o
o
o
1 , 1 , 1 1 , 1 1.
log
)
(
.
log
log
)
(
1
1
δ δβ
β
β
δ δ(Applying the Mean Value Theorem for integrals)
( )
( )
( )
1
(
1
)
log
)
(
).
1
(
1
1
o
o
n
n
o
O
o
o
+
+
+
+
=
β
Also by the virtue of Riemann Lebesgue theorem and regularity of method of summation, we have
I
3=
o
( )
1
, as∞
→
n
, (4.8)Thus from (4.1), (4.3), (4.6), (4.7) and (4.8), we have
{
(
)
(
)
}
(
1
)
0
,
S
x
f
x
o
a
kn
k
k
n
−
=
=
, as
n
→
∞
, uniformly in E.This completes the proof of the theorem.
Remark: 1 If we put
n
k
n
a
nklog
)
1
(
1
,
+
−
=
,β
(
t
)
=
1
∀
t
,[ , ]
a b
=
E
then we obtain the correspondingresult of Saxena [6].
Remark: 2 The result of Tripathi and Singh [8] is a particular case of our theorem if
n k n k n
Q
q
a
,=
− ,=
=
nk k
n
q
Q
0,
)
(
log
)
(
)
(
t
Q
t
q
t
t
t
tα
λ
β
=
and[ , ]
a b
=
E
.REFERENCES:
[1] Shyam Lal, On the degree of approximation of conjugate of a function belonging to weighted
W L
(
P,
ξ
( )
t
)
class by matrix summability means of conjugate series of a Fourier series, Tamkang J. Math., 11 (2000), 7-16.[2] Shyam Lal and J.K. Kushwaha, Approximation of conjugate of functions belonging to the generalized Lipschitz class by lower triangular matrix means, Int. J. Math. Anal., 3 (2009), 21, 1031-1041.
[3] Shyam Lal and Purnima Yadav, Matrix Summability of the conjugate series of derived Fourier series, Tamkang J. Math., 33 (1) (2002), 35-43.
[4] M.L. Mittal and R. Kumar, On matrix summability of Fourier series and its conjugate Series, Bull. Cal. Math. Soc., 82 (1990), 363-368.
[5] K. Qureshi, On the degree of approximation of a periodic function by almost Nörlund means of its Fourier series, Tamkang J. Math., 12 (1), (1981), 35-38.
[6] Ashok Saxena, On uniform harmonic summability of Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India Part A, 31(1965), 303-310.
[7] E. C. Titchmarsh, Theory of Functions, Second edition, Oxford Press, (1939).
[8] L. M. Tripathi and K. N. Singh, On the uniform Nörlund summability of Fourier Series and its conjugate series, Bull. Cal. Math. Soc., 72(1980), 233-239.
[9] O.Töeplitz,Überallagemeiene lineare Mittel bildunger, P.M.F., 22(1913), 113-119.
[10] A. Zygmund, Trigonometric series, Cambridge University Press, (1977).
