Journal of Inequalities in Pure and
Applied Mathematics
LOCALIZATION OF FACTORED FOURIER SERIES
HÜSEYİN BOR
Department of Mathematics
Erciyes University
38039 Kayseri, Turkey
volume 6, issue 2, article 40,
2005.
Received 20 April, 2005;
accepted 02 May, 2005.
Communicated by: L. Leindler
EMail: bor@erciyes.edu.tr
URL: http://fef.erciyes.edu.tr/math/hbor.htm
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2000
Victoria University
ISSN (electronic): 1443-5756
126-05
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Abstract
In this paper we deal with a main theorem on the local property of | N̄, pn |k
summability of factored Fourier series, which generalizes some known results.
2000 Mathematics Subject Classification: 40D15, 40G99, 42A24, 42B15.
Key words: Absolute summability, Fourier series, Local property.
Localization of Factored Fourier
Series
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
6
Hüseyin Bor
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1.
Introduction
P
Let an be a given infinite series with partial sums (sn ). Let (pn ) be a sequence
of positive numbers such that
(1.1)
Pn =
n
X
pv → ∞ as n → ∞,
(P−i = p−i = 0, i ≥ 1).
v=0
The sequence-to-sequence transformation
tn =
(1.2)
n
1 X
p v sv
Pn v=0
defines the sequence (tn ) of the (N̄ , pn ) means of the sequence (sn ) generated
by the sequence of coefficients (pn ).
P
The series an is said to be summable N̄ , pn k , k ≥ 1, if (see [2])
(1.3)
∞ X
n=1
Pn
pn
k−1
|tn − tn−1 |k < ∞.
In the
special case when pn = 1/(n + 1) for all
values
of n (resp. k = 1),
N̄ , pn summability is the same as |C, 1| (resp. N̄ , pn ) summability. Also if
k
we take k = 1 and pn = 1/(n + 1), summability N̄ , pn k is equivalent to the
summability |R, log n, 1|. A sequence (λn ) is said to be convex if ∆2 λn ≥ 0 for
every positive integer n, where ∆2 λn = ∆λn − ∆λn+1 and ∆λn = λn − λn+1 .
Localization of Factored Fourier
Series
Hüseyin Bor
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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
(1.4)
f (t)dt = 0
−π
and
(1.5)
f (t)∼
∞
X
n=1
(an cos nt + bn sin nt) ≡
∞
X
An (t).
n=1
It is well known that the convergence of the Fourier series at t = x is a local
property of the generating function f (i.e. it depends only on the behaviour of
f in an arbitrarily small neighbourhood of x), and hence the summability of the
Fourier series at t = x by any regular linear summability method is also a local
property of the generating function f .
Localization of Factored Fourier
Series
Hüseyin Bor
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2.
Known Results
Mohanty [4] has demonstrated that the |R, log n, 1| summability of the factored
Fourier series
X
(2.1)
An (t)
log(n + 1)
at t = x, is a local property of the generating function of f , whereas the |C, 1|
summability of this series is not. Matsumoto [3] improved this result by replacing the series (2.1) by
(2.2)
X
An (t)
,
{log log(n + 1)}δ
Localization of Factored Fourier
Series
Hüseyin Bor
δ > 1.
Generalizing the above result Bhatt [1] proved the following theorem.
P −1
Theorem A. If (λn ) is a convex sequence such
that
n λn is convergent,
P
then the summability |R, log n, 1| of the series
An (t)λn log n at a point can
be ensured by a local property.
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3.
The Main Result
The aim of the present paper is to prove a more general theorem which includes
of the above results as special cases. Also it should be noted that the conditions
on the sequence (λn ) in our theorem, are somewhat more general than in the
above theorem.
Now we shall prove the following theorem.
Theorem 3.1.
sequence
P Let k ≥ 1. If (λn ) is a non-negative and non-increasing
such that
pn λn is convergent, then the summability N̄ , pn k of the series
P
An (t)λn Pn at a point is a local property of the generating function f .
Localization of Factored Fourier
Series
Hüseyin Bor
We need the following lemmas for the proof of our theorem.
Lemma
3.2. If (λn ) is a non-negative and non-increasing sequence such that
P
pn λn is convergent, where (pn ) is a sequence of positive
P numbers such that
Pn → ∞ as n → ∞, then Pn λn = O(1) as n → ∞ and Pn ∆λn < ∞.
Pm λm = λ m
pn = O(1)
n=0
m
X
n=0
as
m → ∞.
n=0
Applying the Abel transform to the sum
m
X
pn λn = O(1)
Pn ∆λn =
m
X
n=0
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Proof. Since (λn ) is non-increasing, we have that
m
X
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Pm
n=0
pn λn , we get that
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pn λn − Pm λm+1 .
J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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Since λn ≥ λn+1 , we obtain
m
X
Pn ∆λn ≤ Pm λm +
n=0
m
X
p n λn
n=0
= O(1) + O(1) = O(1)
as
m → ∞.
Lemma 3.3. Let k ≥ 1 and sP
n = O(1). If (λn ) is a non-negative and nonincreasing sequence such that pn λn is convergent, where (pn ) is aPsequence
of positive numbers
an λ n P n
such that Pn → ∞ as n → ∞, then the series
is summable N̄ , pn k .
P
Proof. Let (Tn ) be the sequence of (N̄ , pn ) means of the series
an λ n P n .
Then, by definition, we have
n
n
v
1 X X
1 X
pv
ar λ r P r =
(Pn − Pv−1 )av λv Pv .
Tn =
Pn v=0 r=0
Pn v=0
Then, for n ≥ 1, we have
Tn − Tn−1
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Series
Hüseyin Bor
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n
pn X
Pv−1 Pv av λv .
=
Pn Pn−1 v=1
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By Abel’s transformation, we have
Tn − Tn−1
n−1
n−1
pn X
pn X
=
Pv Pv sv ∆λv −
Pv s v p v λv
Pn Pn−1 v=1
Pn Pn−1 v=1
n−1
pn X
−
Pv pv+1 sv λv+1 + sn pn λn
Pn Pn−1 v=1
= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
say.
By Minkowski’s inequality for k > 1, to complete the proof of Lemma 3.3, it is
sufficient to show that
∞
X
(3.1)
(Pn /pn )k−1 |Tn,r |k < ∞,
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Now, applying Hölder’s inequality with indices k and k 0 , where
k > 1, we get that
n=2
Pn
pn
k−1
≤
m+1
X
n=2
Since
1
k
+
1
k0
= 1 and
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|Tn,1 |k
pn
Pn Pn−1
Hüseyin Bor
for r = 1, 2, 3, 4.
n=1
m+1
X
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( n−1
X
)(
k
|sv | Pv Pv ∆λv
v=1
1
n−1
X
Pn−1
v=1
)k−1
Pv Pv ∆λv
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n−1
X
v=1
Pv Pv ∆λv ≤ Pn−1
n−1
X
v=1
Pv ∆λv ,
J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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it follows by Lemma 3.2 that
1
n−1
X
Pn−1
v=1
Pv Pv ∆λv ≤
n−1
X
Pv ∆λv = O(1)
n → ∞.
as
v=1
Therefore
m+1
X
n=2
Pn
pn
k−1
k
|Tn,1 | = O(1)
= O(1)
= O(1)
m+1
X
n=2
m
X
v=1
m
X
n−1
pn X
|sv |k Pv Pv ∆λv
Pn Pn−1 v=1
k
|sv | Pv Pv ∆λv
Localization of Factored Fourier
Series
m+1
X
pn
P P
n=v+1 n n−1
Pv ∆λv = O(1)
Hüseyin Bor
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as m → ∞,
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v=1
by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. Again
m+1
X
n=2
Pn
pn
k−1
|Tn,2 |k ≤
m+1
X
n=2
pn
Pn Pn−1
= O(1)
m+1
X
v=2
m
X
( n−1
X
)(
|sv |k (Pv λv )k pv
v=1
1
n−1
X
Pn−1
v=1
n−1
pn X
|sv |k (Pv λv )k pv
Pn Pn−1 v=1
m+1
X
pn
= O(1)
|sv | (Pv λv ) pv
P P
v=1
n=v+1 n n−1
k
k
)k−1
pv
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= O(1)
m
X
|sv |k (Pv λv )k
v=1
= O(1)
m
X
pv
Pv
|sv |k (Pv λv )k−1 pv λv
v=1
= O(1)
m
X
pv λv = O(1)
as
m → ∞,
v=1
in view of the hypotheses of Theorem 3.1 and Lemma 3.2. Using the fact that
Pv < Pv+1 , similarly we have that
m
m+1
X Pn k−1
X
k
|Tn,3 | = O(1)
pv+1 λv+1 = O(1) as m → ∞.
pn
n=2
v=1
Finally, we have that
k−1
m
m X
X
Pn
k
|Tn,4 | =
|sn |k (Pn λn )k−1 pn λn
pn
n=1
n=1
= O(1)
m
X
pn λn = O(1)
Localization of Factored Fourier
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Hüseyin Bor
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as m → ∞,
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n=1
by virtue of the hypotheses of the theorem and Lemma 3.2. Therefore, we get
that
k−1
m X
Pn
|Tn,r |k = O(1) as m → ∞, for r = 1, 2, 3, 4.
pn
n=1
This completes the proof of Lemma 3.3.
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In the particular case if we take pn = 1 for all values of n in Lemma 3.3,
then we get the following corollary.
Corollary 3.4. Let k ≥ 1 and and sn P
= O(1). If (λn ) is a non-negative
and
λn is convergent, then the series
P non-increasing sequence such that
nan λn is summable |C, 1|k .
Proof of Theorem 3.1. Since the behaviour of the Fourier series, as far as convergence is concerned, for a particular value of x depends on the behaviour of
the function in the immediate neighbourhood of this point only, hence the truth
of Theorem 3.1 is a consequence of Lemma 3.3. If we take pn = 1 for all values of n in this theorem, then we get a new local property result concerning the
|C, 1|k summability.
Localization of Factored Fourier
Series
Hüseyin Bor
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References
[1] S.N. BHATT, An aspect of local property of |R, log n, 1| summability of the
factored Fourier series, Proc. Nat. Inst. Sci. India, 26 (1960), 69–73.
[2] H. BOR, On two summability methods, Math. Proc. Cambridge Philos
Soc., 97 (1985), 147–149.
[3] K. MATSUMOTO, Local property of the summability |R, log n, 1|, Tôhoku
Math. J. (2), 8 (1956), 114–124.
[4] R. MOHANTY, On the summability |R, log w, 1| of Fourier series, J. London Math. Soc., 25 (1950), 67–72.
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Series
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