Bulletin of the Institute of Mathematics
Academia Sinica (New Series)
Vol. 3 (2008), No. 3, pp. 399-406
FACTORS FOR |N̄ , pn , θn |k SUMMABILITY
OF FOURIER SERIES
BY
HÜSEYİN BOR
Abstract
In the present paper, the author presents a generalization
of some known results on the |N̄ , pn |k summability factors for the
|N̄ , pn , θn |k summability factors. Some new results have also been
obtained.
1. Introduction
P
Let
an be a given infinite series with partial sums (sn ). We denote
P
by tn the n-th (C, 1) mean of the sequence (nan ). A series
an is said to
be summable |C, 1|k , k ≥ 1 , if (see [4, 6])
∞
X
1
|tn |k < ∞.
n
(1)
n=1
Let (pn ) be a sequence of positive numbers such that
Pn =
n
X
pv → ∞
as n → ∞,
(P−i = p−i = 0, i ≥ 1).
(2)
v=0
The sequence-to-sequence transformation
σn =
n
1 X
pv s v
Pn
(3)
v=0
Received August 8, 2007 and in revised form December 11, 2007.
AMS Subject Classification: 40D15, 40F05, 40G99.
Key words and phrases: Absolute summability, Fourier series, summability factors.
399
400
HÜSEYİN BOR
[September
defines the sequence (σn ) of the Riesz mean or simply the (N̄ , pn ) mean of
the sequence (sn ), generated by the sequence of coefficients (pn ) (see [5]).
P
The series
an is said to be summable |N̄ , pn |k , k ≥ 1, if (see [1])
∞
X
(Pn /pn )k−1 |∆σn−1 |k < ∞,
(4)
n=1
where
n
pn X
Pv−1 av ,
=−
Pn Pn−1 v=1
∆σn−1
n ≥ 1.
(5)
In the special case pn = 1 for all values of n |N̄ , pn |k summability is the same
as |C, 1|k summability.
P
Let (θn ) be any sequence of positive real constants. The series
an is
said to be summable |N̄ , pn , θn |k , k ≥ 1, if (see [8])
∞
X
θnk−1 |∆σn−1 |k < ∞.
(6)
n=1
In the special case if we take θn = Ppnn , then |N̄ , pn , θn |k summability reduces
to |N̄ , pn |k summability. Also if we take θn = n and pn = 1 for all values
of n, then we get |C, 1|k summability. Furthermore if we take θn = n, then
|N̄ , pn , θn |k summability reduces to |R, pn |k (see [3]) summability.
Let f (t) be a periodic function with period 2π and integrable (L) over
(−π, π). Without any loss of generality we may assume that the constant
term in the Fourier series of f (t) is zero, so that
Z π
f (t) dt = 0
(7)
−π
and
f (t) ∼
∞
X
(an cos nt + bn sin nt) =
n=1
∞
X
An (t).
n=1
We write
1
ϕ(t) = {f (x + t) + f (x − t)} ,
2
1
ϕ1 (t) =
2
Z
t
ϕ(u) du.
0
(8)
2008]
FACTORS FOR |N̄ , pn , θn |k SUMMABILITY OF FOURIER SERIES
401
2. Known Results
In [7] Mishra has proved two theorems for |N̄ , pn | summability factors.
Later on, Bor [2] has generalized these theorems for |N̄ , pn |k summability
factors in the following forms.
Theorem A. Let (pn ) be a sequence such that
Pn = O(npn )
(9)
Pn ∆pn = O(pn pn+1 ).
(10)
If ϕ1 (t) is of bounded variation in (0, π) and (λn ) is a sequence such that
∞
X
1
|λn |k < ∞
n
n=1
(11)
∞
X
(12)
and
|∆λn | < ∞,
n=1
then the series
P
n λn
An (t) Pnp
is summable N̄ , pn k , k ≥ 1.
n
Theorem B. If the sequences (pn ) and (λn ) satisfy the conditions (9)−(12)
of Theorem A and
Bn ≡
n
X
vav = O(n),
(13)
v=1
then the series
P
n λn
an Pnp
is summable |N̄ , pn |k , k ≥ 1.
n
3. Main results.
The aim of this paper is to generalize Theorem A and Theorem B for
|N̄ , pn , θn |k summability methods.
Now we shall prove the following theorems.
Theorem 1. Let ( θnPpnn ) be a non-increasing sequence. If all conditions
402
HÜSEYİN BOR
[September
of Theorem A are satisfied with the condition (11) replaced by;
∞
X
θnk−1 n−k |λn |k < ∞,
(14)
n=1
then the series
P
n λn
An (t) Pnp
is summable |N̄ , pn , θn |k , k ≥ 1.
n
Theorem 2. If the conditions (9)−(10) and (12)−(14) are satisfied and
P Pn λn
is a non-increasing sequence, then the series
an npn is summable
( θPn pnn )
|N̄ , pn , θn |k , k ≥ 1.
Remark. It should be noted that if we take θn =
Pn
pn
in Theorem 1 and
Theorem 2, then we get Theorem A and Theorem B, respectively. In this
case the condition ( θnPpnn ) which is a non-increasing sequence is automatically
satisfied and condition (14) reduces to condition (11).
We need the following lemmas for the proof of our Theorems.
Lemma 1([7]). If ϕ1 (t) is of bounded variation in (0, π), then
X
vAv (x) = O(n) as
n → ∞.
Lemma 2([2]). If the sequence (pn ) such that conditions (9) and (10)
of Theorem A are satisfied, then
∆
1
n P o
n
=
O
.
pn n 2
n2
4. Proof of Theorem 2.
Let (Tn ) denotes the (N̄ , pn ) mean of the series
Then, by definition, we have
Tn
P
n
v
1 X X
=
pv
ar Pr λr (rpr )−1
Pn v=0 r=0
n
1 X
(Pn − Pv−1 )av Pv λv (vpv )−1 .
=
Pn
v=0
an Pn λn (npn )−1 .
2008]
FACTORS FOR |N̄ , pn , θn |k SUMMABILITY OF FOURIER SERIES
403
Then, for n ≥ 1, we have
Tn − Tn−1
n
pn X
=
Pv−1 av Pv λv (vpv )−1 .
Pn Pn−1
v=1
By Abel’s transformation, we have
Tn − Tn−1 = Bn λn n−2 − pn (Pn Pn−1 )−1
n−1
X
pv Pv Bv λv (v 2 pv )−1
v=1
+pn (Pn Pn−1 )−1
+pn (Pn Pn−1 )−1
n−1
X
v=1
n−1
X
Pv Pv ∆λv Bv (v 2 pv )−1
Pv Bv λv+1 ∆
v=1
= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
n P o
v
v 2 pv
say.
To prove the theorem, by Minkowski’s inequality, it is sufficient to show that
∞
X
θnk−1 |Tn,r |k < ∞,
f or
r = 1, 2, 3, 4.
n=1
Firstly, we have that
m
X
θnk−1 |Tn,1 |k =
m
X
θnk−1 |λn |k |Bn |k n−2k
n=1
n=1
= O(1)
= O(1)
m
X
n=1
m
X
θnk−1 |λn |k nk n−2k
θnk−1 n−k |λn |k
n=1
= O(1)
as
m → ∞,
by (13) and (14).
Now, applying Hölder’s inequality, we have that
m+1
X
n=2
θnk−1 |Tn,2 |k
=
m+1
X
n=2
θnk−1
p k
n
Pn
1
|
n−1
X
k
Pn−1
v=1
pv Pv Bv λv k
|
v 2 pv
404
HÜSEYİN BOR
≤
m+1
X
θnk−1
n=2
p k
1
n
Pn
Pn−1
[September
n−1
X
pv
v=1
n P |B ||λ | ok
v
v
v
2
v pv
n 1 n−1
X ok−1
×
pv
Pn−1
v=1
= O(1)
m
X
k
pv |λv |
= O(1)
= O(1)
= O(1)
v=1
m
X
v=1
m
X
k
pv |λv |
|λv |k
v v
n P ok
v
v
pv
v=1
m
X
n P ok
v
pv
k −2k
n=v+1
n P ok−1
v
pv
m+1
X
−k
θ p k−1 p
n
n n
Pn
Pn Pn−1
θ p k−1 m+1
X
pn
v v
Pv
Pn Pn−1
n=v+1
θ p k−1
v v
v −k
Pv
θvk−1 v −k |λv |k = O(1)
as
m → ∞,
v=1
by (13) and (14). On the other hand, since
n−1
X
Pv |∆λv | ≤ Pn−1
n−1
X
|∆λv | ⇒
v=1
v=1
1
Pn−1
n−1
X
Pv |∆λv | ≤
n−1
X
|∆λv | = O(1),
v=1
v=1
by (12), we have that
m+1
X
θnk−1 |Tn,3 |k ≤
n=2
≤
m+1
X
n=2
m+1
X
θnk−1
p k
θnk−1
p k
n=2
n
Pn
n
Pn
n−1
1 X Pv Pv Bv ∆λv k
k
v 2 pv
Pn−1
v=1
1
Pn−1
n−1
X
Pv ∆λv
v=1
n P |B | ok
v
v
2
v pv
n 1 n−1
ok−1
X
×
Pv |∆λv |
Pn−1
v=1
= O(1)
m
X
v=1
k −2k
|Bv | v
n P ok
v
pv
Pv |∆λv |
n=v+1
pn
×
Pn Pn−1
m
θ p k−1
X
v v
v k v −2k v k |∆λv |
= O(1)
Pv
v=1
m+1
X
θ p k−1
n n
Pn
2008]
FACTORS FOR |N̄ , pn , θn |k SUMMABILITY OF FOURIER SERIES
405
m
θ p k−1 X
1 1
= O(1)
|∆λv |
P1
v=1
= O(1)
m
X
|∆λv | = O(1)
as
m → ∞,
v=1
in view of (9), (12) and (13).
Finally, using the fact that ∆{ vP2 pvv } = O( v12 ) by Lemma 2, we get
m+1
X
θnk−1 |Tn,4 |k
≤
m+1
X
θnk−1
n=2
n=2
= O(1)
= O(1)
m+1
X
n=2
m−1
X
n−1
n P ok
pn k 1 n X
v
Pv |Bv |λv+1 ∆ 2
k
Pn Pn−1
v
p
v
v=1
θnk−1
p k
θnk−1
p k
n=2
n
Pn
n
Pn
n−1
1 nX
k
Pn−1
1
Pn−1
v=1
n−1
X
v=1
pv
ok
Pv
|B
||λ
|
v
v+1
v 2 pv
Pv k
pv |λv+1 |k v −2k
pv
n 1 n−1
X ok−1
pv
×|Bv |k
Pn−1
v=1
= O(1)
m X
Pv k
v=1
pv
k −2k k
pv |λv+1 | v
v
m+1
X
n=v+1
θ p k−1
n n
Pn
pn
×
Pn Pn−1
m θ p k−1
X
Pv k−1 −k
v v
= O(1)
v |λv+1 |k
pv
Pv
v=1
= O(1)
m
X
θvk−1 v −k |λv+1 |k = O(1)
as
m → ∞,
v=1
by (13) and (14). Therefore, we get that
m
X
θnk−1 |Tn,r |k = O(1)
as
m → ∞,
for r = 1, 2, 3, 4.
n=1
This completes the proof of Theorem 2.
Proof of Theorem 1. Theorem 1 is a direct consequence of Theorem
2 and Lemma 1. If we take pn = 1 and θn = n in Theorem 1 and Theorem
406
HÜSEYİN BOR
[September
2, then we get the following corollaries. It should be noted that, in this case
condidtion (14) reduces to condition (11).
Corollary 1. If ϕ1 (t) is of bounded variation in (0, π) and (λn ) is a
sequence such that conditions (11) and (12) are satisfied, then the series
P
An (t)λn , at t = x is summable |C, 1|k , k ≥ 1.
P
Corollary 2. If the conditions (11)−(13) are satisfied, then the series
an λn is summable |C, 1|k , k ≥ 1.
References
1. H. Bor, A note on two summability methods, Proc. Amer. Math. Soc., 98(1986),
81-84.
2. H. Bor, Multipliers for |N̄ , pn |k summability of Fourier series, Bull. Inst. Math.
Acad. Sinica, 17(1989), 285-290.
3. H. Bor, On the relative strength of two absolute summability methods, Proc.
Amer. Math. Soc., 113(1991), 1009-1012.
4. T. M. Flett, On an extension of absolute summability and some theorems of
Littlewood and Paley, Proc. London Math. Soc., 7(1957), 113-141.
5. G. H. Hardy, Divergent Series, Oxford Univ. Press, Oxford, 1949.
6. E. Kogbetliantz, Sur les séries absolument par la méthode des moyannes
arithmétiques, Bull. Sci. Math., 49(1925), 234-256.
7. K. N. Mishra, Multipliers for |N̄ , pn | summability of Fourier series, Bull. Inst.
Math. Acad. Sinica, 14(1986), 431-438.
8. W. T. Sulaiman, On some summability factors of infinite series, Proc. Amer. Math.
Soc., 115(1992) , 313-317.
Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey.
E-mail: [email protected], [email protected]