Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 4(2011), Pages 163-168.
ON LOCAL PROPERTY OF ABSOLUTE WEIGHTED MEAN
SUMMABILITY OF FOURIER SERIES
(COMMUNICATED BY HUSEIN BOR)
W.T. SULAIMAN
Abstract. We improve and generalize a result on a local property of |T |k
summability of factored Fourier series due to Sarıgöl [6].
1. Introduction
Let∑
T be a lower triangular matrix, (sn ) a sequence of the partial sums of the
series
an , then
n
∑
Tn :=
tnv sv .
(1)
A series
∑
v=0
an is said to be summable |T |k , k ≥ 1, if (see [6])
∞
∑
|tnn |
1−k
k
|∆Tn−1 | < ∞.
(2)
n=1
Given any lower triangular matrix T one can associate the matrices T and T̂ , with
entries defined by
tnv =
n
∑
tni , n, i = 0, 1, 2, . . . ,
b
tnv = tnv − tn−1,v
i=v
respectively. With sn =
n
∑
ai λi ,
i=0
tn =
n
∑
v=0
tnv sv =
n
∑
v=0
tnv
v
∑
ai λi =
i=0
n
∑
i=0
ai λi
n
∑
v=i
tnv =
n
∑
tni ai λi
i=0
2000 Mathematics Subject Classification. 40F05, 40D25, 40G99.
Key words and phrases. Local property, absolute summability, Fourier series.
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted October 15, 2011. Published October 18, 2011.
163
(3)
164
W.T. SULAIMAN
Yn
:
= tn − tn−1 =
n
∑
tni ai λi −
i=0
=
n
∑
n−1
∑
tn−1,i ai λi
i=0
t̂ni ai λi as tn−1,n = 0.
(4)
i=0
Recall that t̂nn = tnn . (pn ) is assumed to be positive sequences of numbers such
that
Pn = p0 + p1 + . . . + pn → ∞, as n → ∞ ,
The series
∑
an is said to be summable |R, pn |k , k ≥ 1, if
∞
∑
k
nk−1 |∆zn−1 | < ∞ ,
n=1
where
zn =
The series
∑
n
∑
pi si .
i=0
an is said to be summable N , pn k , k ≥ 1, if (see [2])
)k−1
∞ (
∑
Pn
k
|∆zn−1 | < ∞ .
p
n
n=1
Let f be 2π-periodic function, Lebesgue integrable over [−π, π] . Without loss of
generality, we may assume that the constant term in the Fourier series representation for f is zero and that
∞
∞
∑
∑
f (t) ≈
(an cos nt + bn sin nt) ≡
An (t).
(5)
n=1
n=1
It is known that the convergence of the Fourier series of f at t = x is a local
property of f , that is the convergence depends only on the behavior of f in an
arbitrary small neighborhood of x. Therefore it follows that the summability of the
Fourier series at t = x by any regular summability method is also a local property
of f.
A sequence (λn ) is said to be convex if ∆2 λn ≥ 0, (∆2 λn = ∆ (∆λn )).
Mohanty [5] proved that the |R, log n, 1| summability of factored Fourier series
∑
A(t)
(6)
log(n + 1)
at t = x is a local property of f. Matsumoto [3] improved the previous result by
replacing the series in (6) with
∑
An (t)
, δ > 1.
(7)
δ
(log log(n + 1))
Bhatt [1], in turn, generalized the result of Matsumoto by giving the following result
∑ −1
Theorem 1.1. If (λn ) is convex sequence
n λn is convergent, then
∑ such that
the summability |R, log n, 1| of the series
An (t)λn log n at a point can be ensured
by a local property.
ON LOCAL PROPERTY OF ABSOLUTE WEIGHTED MEAN SUMMABILITY ...
165
Bor[2] introduced the following theorem on the local property of the summability
N , pn of the factored Fourier series, which generalizes most of the above results
k
under more appropriate conditions than those given in them.
Theorem 1.2. Let the positive sequence (pn ) and a sequence (λn ) be such that
∞
∑
∆Xn = O(1/n),
(
)
k
k
n−1 |λn | + |λn+1 | Xnk−1 < ∞,
(8)
(9)
n=1
∞
∑
(
)
Xnk + 1 |∆λn | < ∞,
(10)
n=1
where Xn = (npn )−1 Pn . Then the summability N , pn k , k ≥ 1, of the series
∑
λn Xn An (t) at any point is a local property of f.
Finally, Sarıgöl [6] generalized Bor’s result by giving the following
Theorem 1.3. Suppose that T = (tnv ) is a normal matrix such that
tn−1,v ≥ tnv for n ≥ v + 1,
(11)
t̄n0 = 1, n = 0, 1, . . . ,
(12)
n−1
∑
tvv t̂n,v−1 = O(ann ),
(13)
v=1
∆Xn = O(1/n),
(14)
where Xn = (ntnn ) . If a sequence (λn ) holds for k ≥ 1 and the conditions (9),
∞
∑
(10) are satisfied, then the summability |T |k of the series
λn Xn An (t) at any
−1
n=1
point is a local property of f.
The object of the present paper is to improve and generalize Sarıgöl’s result. In
fact we do the following
(1) The matrix we use is not positive in general.
(2) the condition (10) is replaced by following weaker one
∞
∑
Xn |∆λn | < ∞,
Xn ≥ 1.
(15)
n=1
2. Results
We prove the following
Theorem 2.1. Suppose that T = (tnv ) is a normal matrix satisfying
t̂n,v+1 ≤ |tnn | ,
∞
∑
t̂n,v+1 < ∞,
(16)
(17)
n=v+1
n−1
∑
v=1
|tvv | t̂n,v+1 = O(|tnn |),
(18)
166
W.T. SULAIMAN
n−1
∑
∆v t̂nv = O(|tnn |).
(19)
v=1
−1
Suppose that Xn = (n |tnn |)
and satisfied (14). If a sequence (λn ) holds for
k ≥ 1 and the conditions (9), (15) are also satisfied, then the summability |T |k of
∞
∑
the series
λn Xn An (t) at any point is a local property of f.
n=1
The following lemma is needed for the proof of the theorem
Lemma 2.2. Suppose that the matrix T and the sequence (λn ) satisfying the con∞
∑
ditions of the theorem, and that (sn ) is bounded. Then the series
λn Xn An (t) is
n=1
summable |T |k , k ≥ 1.
∞
∑
Proof. Let (Tn ) be a T-transform of the series
λn Xn An (t). By (4)
n=1
n
∑
∆Tn =
t̂nv λv Xv ,
X0 = 0.
v=1
Via Abel’s transformation, we have
∆Tn
=
n−1
∑
n−1
∑
t̂n,v+1 Xv ∆λv sv +
v=1
t̂n,v+1 λv ∆Xv sv +
v=1
n−1
∑
∆t̂nv λv Xv sv + tnn λn Xn sn
v=1
= Tn1 + Tn2 + Tn3 + Tn4 .
In order the prove the lemma, by Minkowski’s inequality, it is sufficient to show
that
∞
∑
1−k
k
|tnn |
|Tnj | < ∞,
j = 1, 2, 3, 4.
n=1
By Hölder’s inequality,
∞
∑
|tnn |
1−k
|Tn1 |
k
=
n=2
≤
n=2
n−1
k
∑
t̂n,v+1 Xv ∆λv sv ∞
∑
n−1
∑
∞
∑
1−k
|tnn |
v=1
1−k
|tnn |
n=2
= O(1)
∞
∑
∞
∑
|tnn |
1−k
∞
∑
n−1
∑
= O(1)
v=1
= O(1)
∞
∑
v=1
Xv |∆λv |
t̂n,v+1 k Xv |∆λv | |sv |k
v=1
∞
∑
Xv |∆λv |
Xv |∆λv |
v=1
∞
∑
)k−1
v=1
v=1
= O(1)
(n−1
∑
v=1
n=2
= O(1)
t̂n,v+1 k Xv |∆λv | |sv |k
Xv |∆λv |
n=v+1
∞
∑
n=v+1
∞
∑
1−k
t̂n,v+1 k
1−k
t̂n,v+1 k−1 t̂n,v+1 |tnn |
|tnn |
t̂n,v+1 n=v+1
Xv |∆λv | = O(1).
ON LOCAL PROPERTY OF ABSOLUTE WEIGHTED MEAN SUMMABILITY ...
∞
∑
|tnn |
1−k
|Tn2 |
k
=
n=2
≤
n=2
n−1
k
∑
sv t̂n,v+1 λv+1 ∆Xv ∞
∑
n−1
∑
∞
∑
1−k
|tnn |
v=1
1−k
|tnn |
n=2
1−k
k
k
|sv | t̂n,v+1 |tvv |
|λv+1 | |∆Xv |
k
v=1
= O(1)
= O(1)
∞ n−1
∑
∑
n=2 v=1
∞
∑
)k−1
|tvv | t̂n,v+1
v=1
1−k
k
∞
∑
t̂n,v+1 k
|∆Xv | |λv+1 |
v=1
= O(1)
(n−1
∑
k
1−k
k
k
|sv | t̂n,v+1 |tvv |
|λv+1 | |∆Xv |
|tvv |
∞
∑
167
n=v+1
k
|λv+1 | |tvv |
1−k
n−k
v=1
= O(1)
∞
∑
v −1 |λv+1 | Xvk−1 = O(1).
k
v=1
∞
∑
1−k
|tnn |
k
|Tn3 |
=
n=2
≤
n=2
n−1
k
∑
sv ∆t̂nv λv Xv ∞
∑
n−1
∑
∞
∑
|tnn |
1−k
v=1
|tnn |
1−k
n=2
k
|sv | ∆v t̂nv |λv | Xvk
k
(n−1
)k−1
∑
∆v t̂nv v=1
= O(1)
= O(1)
= O(1)
∞
∑
n=2
∞
∑
v=1
∞
∑
|tnn |
v=1
1−k
|tnn |
k−1
n−1
∑
k
k
|sv | ∆v t̂nv |λv | Xvk
v=1
k
∞
∑
∆v t̂nv k
|sv | |λv | Xvk
n=v+1
k
|λv | Xvk t̂vv v=1
= O(1)
∞
∑
v −1 |λv | Xvk−1 = O(1).
k
v=1
∞
∑
n=2
1−k
|tnn |
k
|Tn4 |
=
∞
∑
1−k
|tnn |
sn t̂nn λn Xn k
n=2
= O(1)
= O(1)
∞
∑
n=1
∞
∑
|tnn |
1−k
k
k
|tnn | |λn | Xnk
n−1 |λn | Xnk−1 = O(1).
k
n=1
This completes the proof of the lemma.
Remark 2.3. It may be mentioned that whenever T is positive , then conditions
(16)-(19) are replaced by conditions (11)-(13).
168
W.T. SULAIMAN
Proof of Theorem 2.1. Since the convergence of the Fourier series at a point
is a local property of its generating function f , the theorem follows from [7, chapter
II, formula (7.1)] and lemma 2.2.
Corollary 2.4. Suppose that T = (tnv ) is a positive normal matrix. Suppose that
Xn = (n |tnn |)−1 and satisfied (14). If a sequence (λn ) holds for k ≥ 1 and the
conditions (9), (15) are also satisfied, then the summability |R, pn |k of the series
∞
∑
λn Xn An (t) at any point is a local property of f.
n=1
Proof. The proof follows from theorem 2.1 by putting
pv
pn Pv−1
−pn pv
tnv =
, t̂nv =
, ∆v t̂nv =
.
Pn
Pn Pn−1
Pn Pn−1
It is not difficult to check that the conditions (16)-(19) are all satisfied.
References
[1] S. N. Bhatt, An aspect of local property of |R, log n, 1| summability of the factored Fourier
series, Proc. Nat. Inst. India 26 (1968) 69–73.
[2] H. Bor, On the Local property of |N, pn |k summability of the factored Fourier series, J. Math.
Anal. Appl. 163 (1992) 220-226.
[3] K. Matsumoto, Local property of the summability |R, λn , 1| , Thoku Math. J. 2 (8) (1965)
114-124.
[4] S. M. Mazhar, On the summability factors of infinite series, Publ. Math. Debrecen 13 (1956)
229-236.
[5] R. Mohanty, On the summability |R, log w, 1| of a Fourier series, J. London . Soc. 25 (1950)
229-272.
[6] M.A. Sarıgöl, On the local property of |A|k summability of factored Fourier series, J. Math.
Anal. Appl. 188 (1994), 118-127.
[7] A. Zygmund, Trigonometric series, vol 1. Cambridge Univ. Press Canbridge. (1959) .
Waad Sulaiman
Department of Computer Engineering
College of Engineering
University of Mosul, Iraq.
E-mail address: waadsulaiman@hotmail.com