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Journal of Inequalities in Pure and
Applied Mathematics
LOCALIZATION OF FACTORED FOURIER SERIES
volume 6, issue 2, article 40,
HÜSEY˙N BOR
I 2005.
Department of Mathematics Received 20 April, 2005;
Erciyes University accepted 02 May, 2005.
38039 Kayseri, Turkey Communicated by: L. Leindler
EMail: bor@erciyes.edu.tr
URL: http://fef.erciyes.edu.tr/math/hbor.htm
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c 2000 Victoria University
ISSN (electronic): 1443-5756 Quit
126-05
Abstract
¯
In this paper we deal with a main theorem on the local property of | N, 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
Contents Series
Hüseyin Bor
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Title Page
References
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J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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1. Introduction
Let an be a given infinite series with partial sums (sn ). Let (pn ) be a sequence
of positive numbers such that
n
(1.1) Pn = pv → ∞ as n → ∞, (P−i = p−i = 0, i ≥ 1).
v=0
The sequence-to-sequence transformation
Localization of Factored Fourier
n Series
1
(1.2) tn = p v sv Hüseyin Bor
Pn v=0
¯
defines the sequence (tn ) of the (N , pn ) means of the sequence (sn ) generated Title Page
by the sequence of coefficients (pn ). Contents
¯
The series an is said to be summable N , pn k , k ≥ 1, if (see [2])
∞ k−1
Pn
(1.3) |tn − tn−1 |k < ∞.
pn
n=1 Go Back
In the special case when pn = 1/(n + 1) for all values of n (resp. k = 1), Close
¯
¯ , pn summability is the same as |C, 1| (resp. N , pn ) summability. Also if
N Quit
k
¯
we take k = 1 and pn = 1/(n + 1), summability N , pn k is equivalent to the
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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 . J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
http://jipam.vu.edu.au
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
π
(1.4) f (t)dt = 0
−π
and
∞ ∞
Localization of Factored Fourier
(1.5) f (t)∼ (an cos nt + bn sin nt) ≡ An (t). Series
n=1 n=1
Hüseyin Bor
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 Title Page
f in an arbitrarily small neighbourhood of x), and hence the summability of the
Contents
Fourier series at t = x by any regular linear summability method is also a local
property of the generating function f .
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J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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2. Known Results
Mohanty [4] has demonstrated that the |R, log n, 1| summability of the factored
Fourier series
An (t)
(2.1)
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 replac- Localization of Factored Fourier
Series
ing the series (2.1) by
Hüseyin Bor
An (t)
(2.2) , δ > 1.
{log log(n + 1)}δ
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Generalizing the above result Bhatt [1] proved the following theorem. Contents
−1
Theorem A. If (λn ) is a convex sequence such that n λn is convergent,
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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J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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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. Let k ≥ 1. If (λn ) is a non-negative and non-increasing sequence
such that ¯
pn λn is convergent, then the summability N , pn k of the series Localization of Factored Fourier
Series
An (t)λn Pn at a point is a local property of the generating function f .
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 Title Page
pn λn is convergent, where (pn ) is a sequence of positive numbers such that
Contents
Pn → ∞ as n → ∞, then Pn λn = O(1) as n → ∞ and Pn ∆λn < ∞.
Proof. Since (λn ) is non-increasing, we have that
m m
Pm λm = λ m pn = O(1) pn λn = O(1) as m → ∞. Go Back
n=0 n=0 Close
m
Applying the Abel transform to the sum n=0 pn λn , we get that Quit
m m Page 6 of 12
Pn ∆λn = pn λn − Pm λm+1 .
n=0 n=0 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
http://jipam.vu.edu.au
Since λn ≥ λn+1 , we obtain
m m
Pn ∆λn ≤ Pm λm + p n λn
n=0 n=0
= O(1) + O(1) = O(1) as m → ∞.
Lemma 3.3. Let k ≥ 1 and sn = O(1). If (λn ) is a non-negative and non- Localization of Factored Fourier
increasing sequence such that pn λn is convergent, where (pn ) is a sequence Series
of positive numbers such that Pn → ∞ as n → ∞, then the series an λ n P n Hüseyin Bor
¯
is summable N , pn k .
¯
Proof. Let (Tn ) be the sequence of (N , pn ) means of the series an λ n P n . Title Page
Then, by definition, we have Contents
n v n
1 1
Tn = pv ar λ r P r = (Pn − Pv−1 )av λv Pv .
Pn v=0 r=0
Pn v=0
Then, for n ≥ 1, we have Go Back
n Close
pn
Tn − Tn−1 = Pv−1 Pv av λv . Quit
Pn Pn−1 v=1
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J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
http://jipam.vu.edu.au
By Abel’s transformation, we have
n−1 n−1
pn pn
Tn − Tn−1 = Pv Pv sv ∆λv − Pv s v p v λv
Pn Pn−1 v=1
Pn Pn−1 v=1
n−1
pn
− Pv pv+1 sv λv+1 + sn pn λn
Pn Pn−1 v=1
= Tn,1 + Tn,2 + Tn,3 + Tn,4 , say.
Localization of Factored Fourier
By Minkowski’s inequality for k > 1, to complete the proof of Lemma 3.3, it is Series
sufficient to show that Hüseyin Bor
∞
(3.1) (Pn /pn )k−1 |Tn,r |k < ∞, for r = 1, 2, 3, 4.
n=1 Title Page
Now, applying Hölder’s inequality with indices k and k , where 1
+ 1
= 1 and Contents
k k
k > 1, we get that
m+1 k−1
Pn
|Tn,1 |k
n=2
pn Go Back
m+1 n−1 n−1 k−1 Close
pn k 1
≤ |sv | Pv Pv ∆λv Pv Pv ∆λv . Quit
n=2
Pn Pn−1 v=1
Pn−1 v=1
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Since
n−1 n−1
Pv Pv ∆λv ≤ Pn−1 Pv ∆λv , J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
http://jipam.vu.edu.au
v=1 v=1
it follows by Lemma 3.2 that
n−1 n−1
1
Pv Pv ∆λv ≤ Pv ∆λv = O(1) as n → ∞.
Pn−1 v=1 v=1
Therefore
m+1 k−1 m+1 n−1
Pn k pn
|Tn,1 | = O(1) |sv |k Pv Pv ∆λv
n=2
pn n=2
Pn Pn−1 v=1
Localization of Factored Fourier
Series
m m+1
k pn
= O(1) |sv | Pv Pv ∆λv Hüseyin Bor
v=1
P P
n=v+1 n n−1
m
Title Page
= O(1) Pv ∆λv = O(1) as m → ∞,
v=1 Contents
by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. Again
m+1 k−1 m+1 n−1 n−1 k−1
Pn k pn k k 1
|Tn,2 | ≤ |sv | (Pv λv ) pv pv Go Back
n=2
pn n=2
Pn Pn−1 v=1
Pn−1 v=1
m+1 n−1 Close
pn k k
= O(1) |sv | (Pv λv ) pv Quit
v=2
Pn Pn−1 v=1
m m+1 Page 9 of 12
k pn
k
= O(1) |sv | (Pv λv ) pv
v=1
P P
n=v+1 n n−1 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
http://jipam.vu.edu.au
m
pv
= O(1) |sv |k (Pv λv )k
v=1
Pv
m
= O(1) |sv |k (Pv λv )k−1 pv λv
v=1
m
= O(1) 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 Localization of Factored Fourier
Series
Pv < Pv+1 , similarly we have that
m+1 k−1 m Hüseyin Bor
Pn k
|Tn,3 | = O(1) pv+1 λv+1 = O(1) as m → ∞.
n=2
pn v=1
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Finally, we have that
Contents
m k−1 m
Pn
|Tn,4 |k = |sn |k (Pn λn )k−1 pn λn
n=1
pn n=1
m
= O(1) pn λn = O(1) as m → ∞, Go Back
n=1
Close
by virtue of the hypotheses of the theorem and Lemma 3.2. Therefore, we get
that Quit
m k−1
Pn Page 10 of 12
|Tn,r |k = O(1) as m → ∞, for r = 1, 2, 3, 4.
n=1
pn
J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
This completes the proof of Lemma 3.3. http://jipam.vu.edu.au
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 = O(1). If (λn ) is a non-negative
and non-increasing sequence such that λn is convergent, then the series
nan λn is summable |C, 1|k .
Proof of Theorem 3.1. Since the behaviour of the Fourier series, as far as con-
vergence 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 Localization of Factored Fourier
Series
of Theorem 3.1 is a consequence of Lemma 3.3. If we take pn = 1 for all val-
ues of n in this theorem, then we get a new local property result concerning the Hüseyin Bor
|C, 1|k summability.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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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. Localization of Factored Fourier
Series
[4] R. MOHANTY, On the summability |R, log w, 1| of Fourier series, J. Lon-
don Math. Soc., 25 (1950), 67–72. Hüseyin Bor
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