Localization of Factored Fourier Series by linzhengnd

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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


                                                        Abstract
                                                       Contents




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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
                                                                                                                          Contents




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                                                                                                          J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
                                                                                                              http://jipam.vu.edu.au
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
                                                                                                  Page 3 of 12
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
                                                                                         http://jipam.vu.edu.au
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
                                                                                                    Title Page
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
                                                                                         http://jipam.vu.edu.au
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
                                                                                                                      Page 8 of 12
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
                                                                                                              Title Page
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.
                                                                                                  Title Page
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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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                                                                                  J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005
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