Filomat 30:10 (2016), 2803–2807
DOI 10.2298/FIL1610803B
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
On Absolute Weighted Mean Summability
of Infinite Series and Fourier Series
Hüseyin Bor
P. O. Box 121, TR-06502 Bahçelievler, Ankara, Turkey
Abstract. In this paper, firstly we proved a known theorem dealing with absolute weighted mean summability of infinite series under weaker conditions and then we obtained an application of it to the Fourier
series. Some new results are also deduced.
1. Introduction
P
Let an be a given infinite series with the partial sums (sn ). By uαn and tαn we denote the nth Cesàro
means of order α, with α > −1, of the sequences (sn ) and (nan ), respectively, that is (see [3])
uαn =
n
1 X α−1
A sv
Aαn v=0 n−v
and tαn =
n
1 X α−1
A vav ,
Aαn v=0 n−v
(1)
where
(α + 1)(α + 2)....(α + n)
= O(nα ), Aα−n = 0 f or n > 0.
n!
P
The series an is said to be summable |C, α|k , k ≥ 1, if (see [5], [7])
Aαn =
∞
X
n=1
∞
k X
1 α k
t < ∞.
nk−1 uαn − uαn−1 =
n n
(2)
(3)
n=1
If we take α = 1, then |C, α|k summability reduces to |C, 1|k summability. Let (pn ) be a sequence of positive
real numbers such that
Pn =
n
X
pv → ∞ as
n → ∞,
P−i = p−i = 0,
i≥1 .
(4)
v=0
2010 Mathematics Subject Classification. Primary 26D15; Secondary 40D15, 40F05, 40G99, 42A24, 46A45
Keywords. Weighted mean; summability factors; infinite series; Fourier series; Hölder inequality; Minkowski inequality; sequence
space
Received: 04 August 2014; Accepted: 27 December 2014
Communicated by Dragan S. Djordjević
Email address: [email protected] (Hüseyin Bor)
H. Bor / Filomat 30:10 (2016), 2803–2807
2804
The sequence-to-sequence transformation
wn =
n
1 X
pv sv
Pn v=0
(5)
defines the sequence (wn ) of the N̄, pn mean of the sequence
(sn ) generated by the sequence of coefficients
P
(pn ) (see [6]). The series an is said to be summable N̄, pn k , k ≥ 1, if (see [1])
!k−1
∞
X
Pn
|wn − wn−1 |k < ∞.
pn
n=1
In the special case when pn = 1 for all values of n (resp. k = 1), |N̄, pn |k summability is the same as |C, 1|k ,
(resp.
A sequence (λn ) is said to be of bounded variation, denoted by (λn ) ∈ BV, if
P∞ |N̄, pn |)Psummability.
∞
=
|
λ
−
λ
|∆λ
|
n
n
n+1 |< ∞.
n=1
n=1
2. Known Result
The following theorem is known dealing with |N̄, pn |k summability factors of infinite series.
Theorem 2.1 ([2]) Let (pn ) be a sequence of positive numbers such that
Pn = O(npn )
as
n → ∞.
(6)
Let (Xn ) be a positive monotonic nondecreasing sequence. If the sequences (Xn ), (λn ), and (pn ) satisfy the
conditions
λm Xm = O(1)
m
X
as
m → ∞,
(7)
nXn |∆2 λn | = O(1),
(8)
n=1
m
X
pn k
|tn | = O(Xm ) as
Pn
m → ∞,
(9)
n=1
then the series
P
an λn is summable |N̄, pn |k , k ≥ 1.
3. The Main Result
The aim of this paper is to prove Theorem 2.1 under weaker conditions. Now we shall prove the
following theorem.
Theorem 3.1 Let (Xn ) be a positive monotonic nondecreasing sequence. If the sequences (Xn ), (λn ), and (pn )
satisfy the conditions (6)-(8) and
m
X
pn |tn |k
= O(Xm ) as
Pn Xn k−1
m → ∞,
(10)
n=1
P
then the series an λn is summable |N̄, pn |k , k ≥ 1.
Remark 3. 2 It should be noted that condition (10) is reduced to the condition (9), when k=1. When k > 1,
condition (10) is weaker than condition (9) but the converse is not true. As in [9] we can show that if (9) is
satisfied, then we get that
m
m
X
pn |tn |k
1 X pn k
=
O(
)
|tn | = O(Xm ).
Pn Xn k−1
Pn
X1k−1
n=1
n=1
H. Bor / Filomat 30:10 (2016), 2803–2807
2805
If (10) is satisfied, then for k > 1 we obtain that
m
m
m
X
X
pn |tn |k
pn k X k−1 pn |tn |k
k−1
k
=
O(X
)
= O(Xm
) , O(Xm ).
|tn | =
Xn
m
k−1
Pn
Pn Xn
Pn Xn k−1
n=1
n=1
n=1
We need the following lemma for the proof of our theorem.
Lemma 3. 3 ([2]) Under the conditions of Theorem 3. 1, we have that
∞
X
Xn |∆λn | < ∞,
(11)
n=1
nXn |∆λn | = O(1)
as
n → ∞.
(12)
4. Proof of Theorem 3.1 Let (Tn ) be the sequence of (N̄, pn ) mean of the series
we have
Tn =
P
an λn . Then, by definition,
n
v
n
1 X X
1 X
pv
ar λr =
(Pn − Pv−1 )av λv .
Pn v=0 r=0
Pn v=0
(13)
Then, for n ≥ 1, we get
Tn − Tn−1 =
n
pn X Pv−1 λv
vav .
Pn Pn−1
v
(14)
v=1
Applying Abel’s transformation to the right-hand side of (14), we have
Tn − Tn−1
=
n−1
v
n
pn X Pv−1 λv X
pn λn X
∆
rar +
vav
Pn Pn−1
v
nPn
v=1
=
pn
(n + 1)pn tn λn
−
nPn
Pn Pn−1
+
=
r=1
pn
Pn Pn−1
n−1
X
Pv ∆λv tv
v=1
n−1
X
v=1
r=1
pv tv λv
v+1
v
n−1
pn X
v+1
1
+
Pv λv+1 tv
v
Pn Pn−1
v
v=1
Tn,1 + Tn,2 + Tn,3 + Tn,4 .
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
!k−1
∞
X
k
Pn
Tn,r < ∞,
pn
f or
r = 1, 2, 3, 4.
n=1
Firstly, we have that
!k−1
m
X
Pn
|Tn,1 |k
pn
=
O(1)
n=1
m
X
m
|λn |k−1 |λn |
n=1
=
O(1)
m−1
X
n=1
=
O(1)
m−1
X
n=1
X
pn k
pn |tn |k
|tn | = O(1)
|λn |
Pn
Pn Xn k−1
n=1
∆|λn |
n
X
v=1
m
X pn |t |k
pv |tv |k
n
+ O(1)|λm |
k−1
Pv Xv
Pn Xn k−1
|∆λn |Xn + O(1)|λm |Xm = O(1)
n=1
as
m → ∞,
H. Bor / Filomat 30:10 (2016), 2803–2807
2806
by virtue of the hypotheses of Theorem 3. 1 and Lemma 3. 3. Also, as in Tn,1 , we have that
!k−1
m+1
X
Pn
|Tn,2 |k
p
n
n=2
=
O(1)
m+1
X
pn
Pn Pn−1
n=2
=
O(1)
m
X
 n−1
 
k−1
n−1
X
  1 X

k
k



pv 
pv |tv | |λv |  × 

Pn−1
|λv |k−1 |λv |pv |tv |k
v=1
=
O(1)
v=1
v=1
m+1
X
n=v+1
m
X
pn
Pn Pn−1
k
|λv |
v=1
pv |tv |
= O(1)
Pv Xv k−1
as
m → ∞.
Again, by using (6), we get that
!k−1
m+1
X
k
Pn
Tn,3 pn
n=2

k
n−1




X

=
P
|∆λ
||t
|


v
v
v


k


P
P
n
n−1 v=1
n=2

k
m+1
n−1
X

pn X


vp
|∆λ
||t
|
= O(1)
v
v
v

P Pk 
m+1
X
pn
n n−1
n=2
= O(1)
m+1
X
pn
Pn Pn−1
n=2
= O(1)
m
X
v=1
 n−1
k−1
 
n−1
X
X

 
 (v|∆λ |)k p |t |k  ×  1
pv 
v
v v 

  Pn−1
v=1
v=1
(v|∆λv |)k−1 v|∆λv |pv |tv |k
v=1
= O(1)
m
X
v=1
= O(1)
m−1
X
n=v+1
= O(1)
pn
Pn Pn−1
pv |tv |k
v|∆λv |
Pv Xv k−1
∆ (v|∆λv |)
v=1
m−1
X
m+1
X
m
v
X
X
pv |tv |k
pr |tr |k
+
O(1)m|∆λ
|
m
k−1
Pr Xr
Pv Xv k−1
v=1
r=1
|∆ (v|∆λv |)| Xv + O(1)m|∆λm |Xm
v=1
= O(1)
m−1
X
vXv |∆2 λv | + O(1)
v=1
= O(1)
m−1
X
Xv |∆λv | + O(1)m|∆λm |Xm
v=1
as
m → ∞,
by virtue of the hypotheses of Theorem 3. 1 and and Lemma 3. 3. Finally, by using (6), we have that
!k−1
m+1
X
k
Pn
Tn,4 pn
n=2
= O(1)
= O(1)
m+1
X
pn
n=2
Pn Pkn−1
 n−1
k
X


|λv+1 |pv |tv |

pn
Pn Pn−1
 
k−1
 n−1
n−1
X
  1 X


|λv+1 |k pv |tv |k  × 
pv 

Pn−1
m+1
X
n=2
= O(1)
m
X
v=1
= O(1)
m
X
v=1
v=1
v=1
|λv+1 |k−1 |λv+1 |pv |tv |k
v=1
m+1
X
n=v+1
pv |tv |k
|λv+1 |
= O(1)
Pv Xv k−1
pn
Pn Pn−1
as
m → ∞.
H. Bor / Filomat 30:10 (2016), 2803–2807
2807
This completes the proof of Theorem 3. 1. It should be noted that if we take pn = 1 for all values of n, then
we get the known result of Mazhar dealing with |C, 1|k summability factors of infinite series under weaker
conditions (see [8]).
5. Let f (t) be a periodic function with period 2π and integrable (L) over (−π, π). Write
∞
f (x) ∼
∞
X
X
1
a0 +
(an cos nx + bn sin nx) =
Cn (x)
2
n=0
n=1
Rt
φ(t) = 12 { f (x + t) + f (x − t)}, and φα (t) = tαα 0 (t − u)α−1 φ(u)du, (α > 0).
It is well know that if φ1 (t) ∈ BV(0, π), then tn (x) = O(1), where tn (x) is the (C, 1) mean of the sequence
(nCn (x)) (see [4]). Using this fact, we get the following main result dealing with Fourier series.
Theorem 5. 1 If φ1 (t) ∈ BV(0, π), and the sequences
(pn ), (λn ), and (Xn ) satisfy the conditions of Theorem
P
3. 1, then the series series Cn (x)λn is summable N̄, pn k , k ≥ 1.
If we take pn = 1 for all values of n, then we obtain a new result dealing with |C, 1|k summability factors of
Fourier series.
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