Abstract: In this paper we study A-statistical summability of conjugate Fourier series, derived Fourier series and Walsh-Fourier series. At the end of the paper it is shown that A-statistical summability version is stronger than A-summability

A Alotaibi

M Mursaleen

CiteSeerX

Appl. Math. Inf. Sci. 6, No. 3, 535-538 (2012) 535
Applied Mathematics & Information Sciences
An International Journal
c© 2012 NSP
Natural Sciences Publishing Cor.
A-Statistical Summability of Fourier Series and
Walsh-Fourier Series
A. Alotaibi1 and M. Mursaleen2
1 Department of Mathematics, King Abdul Aziz University, P.O.Box 80203 Jeddah, KSA
2 Department of Mathematics, Aligarh Muslim University, Aligarh, India
Received: May 7, 2011; Revised Nov. 21, 2011; Accepted 16 Dec. 2011
Published online: 1 Sep. 2012
Abstract: In this paper we study A-statistical summability of conjugate Fourier series, derived Fourier series and Walsh-Fourier series.
At the end of the paper it is shown that A-statistical summability version is stronger than A-summability.
Keywords: Statistical convergence; A-statistical convergence; Fourier series; Walsh-Fourier series.
1. Introduction
Let K = {ki} be an index set and let ϕK be the character-
istic sequence of K, i.e. ϕK = (ϕKj ) with
ϕKj =
{
1, for j ∈ K;
0, otherwise.
If ϕK is C1-summable then the limit
lim
n
n∑
j=1
ϕKj
is called the asymptotic density of K and is denoted by
δ(K). A sequence x = (xk) of real numbers is said to
be statistically convergent to L if δ(Kε) = 0 for every
ε > 0, where Kε := {k ∈ N : |xk − L| ≥ ε} ( cf. [4],
[7]). In this case L is called the st-limit of x.
Let c and l∞ denote the set of all convergent and bounded
sequences, respectively. Let X and Y be two sequence
spaces and A = (ank)∞n;k=1 be an infinite matrix of real
or complex numbers. We write Ax = (An(x)) provided
An(x) =
∑
k ankxk converges for each n. A sequence
x = (xk) is said to be A-summable to L if limnAn(x) =
L. If x = (xk) ∈ X implies that Ax ∈ Y , then we say
that A defines a matrix transformation from X into Y and
by (X,Y ) we denote the class of such matrices. If X and
Y are equipped with the limits X-lim and Y -lim, respec-
tively, A ∈ (X,Y ) and Y -limnAn(x) = X-limk xk for
all x ∈X, then we say thatA is a regular map fromX into
Y and in this case we write A ∈ (X,Y )reg. The matrices
A ∈ (c, c)reg are called regular.
For a non-negative regular matrix A = (ank)∞n,k=1,
following Freedman and Sember [6], an index set K =
{ki} is said to have A-density
δA(K) = lim
n
Anϕ
K
if AϕK ∈ c. Thus
δA(K) = lim
n
∑
k∈K
ank = lim
n
∑
i
an,ki .
A sequence x is said to beA-statistically convergent
to L if δA(Kε) = 0 for every ε > 0. In this case we write
stA-limx = L, and xk
stA−→ L. By the symbol stA we
denote the set of all A-statistically convergent sequences.
Note that c ⊂ stA.
The following characterization is due to Kolk [9, Corol-
lary 4] which will be needed in our results.
Lemma 11B = (bnk)∞n,k=1 ∈ (c, stA∩ l∞)reg if and only
if
(i)‖ B ‖= ∑k | bnk |< ∞, and there exists N = {ni}
so that δA(N) = 1 and
(ii)limi bnik = 0 (k ∈ N ), i.e. stA-lim bnk = 0 for each
k,
∗ Corresponding author: e-mail: aalotaibi@kau.edu.sa
c© 2012 NSP
Natural Sciences Publishing Cor.
536 A. Alotaibi and M. Mursaleen: A-Statistical Summability of Fourier Series and Walsh-Fourier Series
(iii)limi
∑
k bnik = 1, i.e. stA-lim
∑
k bnk = 1.
Zygmund proved some theorems on the statistical con-
vergence of Fourier series in the first edition of his book
( [12], pp 181-188 ). Recently, A-statistical convergence
have been used for some approximating operators in [1],
[3] and [8]; and for Fourier integrals [10]. In this paper, we
prove some results on statistical summability of conjugate
Fourier series, derived Fourier series and Walsh-Fourier
series.
2. Preliminaries
2.1.Let f be L-integrable and periodic with period 2π, and
let the Fourier series of f be
1
a0
+
∞∑
k=1
(ak cos kx+ bk sin kx) . (2.1)
Then the series conjugate to it is
∞∑
k=1
(bk cos kx− ak sin kx) , (2.2)
with partial sum S̃n(x); and the derived series is
∞∑
k=1
k (bk cos kx− ak sin kx) , (2.3)
with partial sum S′n(x). We write
ψx(t) = ψ(f, t){
= f(x+ t) − f(x− t) , for 0 < t ≤ π;
g(x) , for t = 0 ; (2.4)
and
βx(t) =
ψx(t)
4 sin 12 t
, (2.5)
where g(x) = f(x+ 0) − f(x− 0).
We shall need the following lemma which is statisti-
cal version of the Banach Weak Convergence Theorem
[2].
Lemma 21stA-limn→∞
∫ π
0
gndhx = 0 for all hx ∈
BV [0, π], if and only if ‖ gn ‖< ∞ for all n and stA-
limn→∞ gn = 0 ; where BV [0, π] denotes the set of
all functions of bounded variations on [0, π].
Parallel version also holds if [0, π] is replaced by [0, 1].
2.2.Let us define a sequence of functions
h0(x), h1(x), ......, hn(x)
which satisfy the following conditions:
h0(x) =
{
1 , for 0 ≤ x ≤ 12 ;−1 , for 12 ≤ x < 1 ;
(2.6)
h0(x + 1) = h0(x) and hn(x) = h0(2nx), n =
1, 2, ......The functions hn(x) are called theRademacher′s
functions.
The Walsh functions are defined by φ0(x) = 1,
φn(x) = hn1(x)hn2(x).......hnr (x), 0 ≤ x ≤ 1,
(2.7)
for n = 2n1 +2n2 + ......2nr ; where the integers ni are
uniquely determined by ni+1 < ni.
Let f be L-integrable and periodic with period 1, and
let the Walsh-Fourier series of f be
∞∑
n=1
cnφn(x), (2.8)
where
cn =
∫ 1
0
f(x)φn(x)dx, (2.9)
are called the Walsh-Fourier coefficients of f.
Let us recall some basic properties of Walsh functions
(see [4]).
For each fixed x ∈ [0, 1) and for all t ∈ [0, 1)
(i)φn(x+̇t) = φn(x)φn(t),
(ii)
∫ 1
0
f(x+̇t)dt =
∫ 1
0
f(t)dt, and
(iii)
∫ 1
0
f(t)φn(x+̇t)dt =
∫ 1
0
f(x+̇t)φn(t)dt;
where +̇ denotes the operation in the dyadic group,
the set of all sequences s = (sn), sn = 0, 1 for n =
1, 2, ....., is addition modulo 2 in each coordinate.
Let for x ∈ [0, 1),
Jk(x) =
∫ x
0
φk(t)dt, k = 0, 1, 2, ... (2.10)
It is easy to see that Jk(x) = 0 for x = 0, 1.
3. Main results
We prove the following theorems:
Theorem 31LetB = (bnk)∞n,k=1 ∈ (c, stA∩l∞)reg . Then
for every x ∈ [−π, π] for which βx(t) ∈ BV [0, π],
stA- lim
m
∞∑
n=1
bmnS
′
n(x) = βx(0+) (3.1.1)
if and only if
stA- lim
m
∞∑
n=1
bmn sin(n+
1
2
)t = 0 for all t ∈ [0, π].
(3.1.2)
Theorem 32Let B ∈ (c, stA ∩ l∞)reg. Then for every x ∈
[−π, π] for which βx(t) ∈ BV [0, π],
stA- lim
m
∞∑
n=1
nbmnS̃n(x) =
1
π
g(x) (3.2.1)
c© 2012 NSP
Natural Sciences Publishing Cor.
Appl. Math. Inf. Sci. 6, No. 3, 535-538 (2012) / www.naturalspublishing.com/Journals.asp 537
if and only if
stA- lim
m
∞∑
n=1
bmn cosnt = 0 for all t ∈ [0, π]. (3.2.2)
Theorem 33LetB ∈ (c, stA∩l∞)reg . Let zk(x) = ckφk(x)
for an L-integrable function f ∈ BV [0, 1). Then for every
x ∈ [0, 1)
stA- lim
n
∞∑
k=1
bnkzk(x) = 0 (3.3.1)
if and only if
stA- lim
n
∞∑
k=1
bnkJk(x) = 0, (3.3.2)
where x is a point at which f(x) is of bounded variation.
4. Proofs
Proof(Proof of Theorem 3.1. ). We have
S′n(x) =
1
π
∫ π
0
ψx(t)
(
n∑
k=1
k sin kt
)
dt
= − 1
π
∫ π
0
ψx(t)
d
dt
(
sin(n+ 12 )t
2 sin t2
)
dt,
= In +
2
π
∫ π
0
sin(n+
1
2
)t dβx(t), (3.1.3)
where
In =
1
π
∫ π
0
βx(t) cos
t
2
(
sin(n+ 12 )t
sin t2
)
dt. (3.1.4)
Then
∞∑
n=1
bmnS
′
n(x) =
∞∑
n=1
bmnIn +
2
π
∫ π
0
Lm(t) dβx(t),
(3.1.5)
where
Lm(t) =
∞∑
n=1
bmn sin(n+
1
2
)t. (3.1.6)
Since βx(t) is of bounded variation on [0, π] and βx(t) →
βx(0+) as t → 0, βx(t) cos t2 has also the same property.
Hence by Jordan’s convergence criterion for Fourier series
In → βx(0+) as n→ ∞.
SinceB ∈ (c, stA∩l∞)reg , by the conditions of Lemma
1.1, we get
stA- lim
m
∞∑
n=1
bmnIn = βx(0+). (3.1.7)
Now, it is enough to show that (3.1.2) holds if and only if
stA- lim
m
∫ π
0
Lm(t) dβx(t) = 0. (3.1.8)
Hence, by Lemma 2.1, it follows that (2.1.8) holds if and
only if
‖ Lm(t) ‖≤M for all m and for all t ∈ [0, π], (3.1.9)
and (3.1.2) holds. Since (3.1.9) is satisfied by Lemma 1.1(i),
it follows that (3.1.8) holds if and only if (3.1.2) holds.
Hence the result follows immediately.
Proof(Proof of Theorem 3.2). We have
We have
S̃n(x) =
1
π
∫ π
0
ψx(t) sinnt dt,
=
g(x)
nπ
+
1
nπ
∫ π
0
cosnt dψx(t).
Therefore
∞∑
n=1
nbmnS̃n(x) =
g(x)
π
∞∑
n=1
bmn +
1
π
∫ π
0
Km(t) dψx(t),
(3.2.3)
where
Km(t) =
∞∑
n=1
bmn cosnt.
Now, taking stA-lim on both sides of (3.2.3) and using
Lemmas 1.1 and 2.1 as in the proof of Theorem 3.1, we
get the required result.
Proof(Proof of Theorem 3.3). We have
zk(x) = ckφk(x) =
∫ 1
0
f(t)φk(t)φk(x)dt,
=
∫ 1
0
f(t)φk(x+̇t)dt =
∫ 1
0
f(x+̇t)φk(t)dt,
where x+̇t belongs to the set Ω of dyadic rationals in
[0, 1), in particular each element of Ω has the form p/2n
for some non-negative integers p and n, 0 ≤ p < 2n.Now,
on integration by parts, we obtain
zk(x) = [f(x+̇t)Jk(t)]10 −
∫ 1
0
Jk(t)df(x+̇t),
= −
∫ 1
0
Jk(t)df(x+̇t), since Jk(x) = 0 for x = 0, 1.
Hence
∞∑
k=1
bnkzk(x) = −
∫ 1
0
Dn(t) dhx(t), (3.3.3)
c© 2012 NSP
Natural Sciences Publishing Cor.
538 A. Alotaibi and M. Mursaleen: A-Statistical Summability of Fourier Series and Walsh-Fourier Series
where
Dn(t) =
∞∑
k=1
bnkJk(t), (3.3.4)
and hx(t) = f(x+̇t).Write, for any t ∈ R, gn = (Dn(t)).
Since B ∈ (c, stA ∩ l∞)reg , it follows by Lemma 1.1
that ‖ gn ‖<∞ for all n, and gn stA−→ 0. Hence by Lemma
2.1, ∫ 1
0
Dn(t) dhx(t)
stA−→ 0.
Now, taking stA-lim in (3.3.3) and (3.3.4) and using
Lemma 2.1, we get the desired result.
Remark. Theorem 3.3 generalizes the result of Mursaleen
[11] and the following example shows that this theorem is
stronger than that of [11].
Example. Let A be a (C, 1) matrix, B = I (identity
matrix). Let us define a sequence u = (uk) by
uk :=
{
1, if k = m2,m ∈ N ;
0, otherwise.
Then u is A-statistically convergent to 0 but not A-
summable. Now write ẑk(x) = (1+uk)zk(x) and Ĵk(x) =
(1 + uk)Jk(x), it is easy to see that Theorem 3.3 holds if
we replace zk(x) by ẑk(x) and Jk(x) by Ĵk(x) but Theo-
rem 3.1 of [11] does not hold.
References
[1] O. Agratini , A-Statistical convergence of a class of integral
operators, Applied Math. & Information Sci. (to appear).
[2] S. Banach, Théorie des operations lineaires, Warszawa,
1932.
[3] O. Duman, M. K. Khan and C. Orhan, A-Statistical con-
vergence of approximating operators, Math. Ineq. Appl.,
6(4)(2003) 689-699.
[4] H. Fast, Sur la convergence statistique, Colloq. Math.,
2(1951) 241-244.
[5] N. J. Fine, On the Walsh-functions, Trans. Amer. Math. Soc.,
65(1949) 372-414.
[6] A. R. Freedman and J. J. Sember, Densities and summabil-
ity, Pacific J. Math., 95(1981) 293-305.
[7] J. A. Fridy, On statistical convergence, Analysis, 5(1985)
301-313.
[8] N. İspir, A-Statistical approximation by Jayasri operators,
Math. Comm., 8(2003) 85-94.
[9] E. Kolk, Matrix maps into the space of statistically conver-
gent bounded sequences, Proc. Estonia Math. Soc. Phys.
Math., 45(1996) 187-192.
[10] F. Moricz, Strong Cesàro summability and statistical limit
of Fourier integrals, Analysis, 25(2005) 79-86.
[11] Mursaleen, Application of infinite matrices to Walsh func-
tions, Demons. Math., 27 (1994) 279-282.
[12] A. Zygmund, Trigonometric Series, 2nd Ed., Cambridge
Univ. Press, 1959.
c© 2012 NSP
Natural Sciences Publishing Cor.


A-Statistical Summability of Fourier Series and Walsh-Fourier

In this paper we study A-statistical summability of conjugate Fourier series, derived Fourier series and Walsh-Fourier series. At the end of the paper it is shown that A-statistical summa- bility version is stronger than A-summability

Alotaibi, Abdullah

Mursaleen, M.

Arab Journals Platform

Appl. Math. Inf. Sci. 6, No. 3, 535-538 (2012) 535Applied Mathematics & Information SciencesAn International Journalc© 2012 NSPNatural Sciences Publishing Cor.A-Statistical Summability of Fourier Series andWalsh-Fourier SeriesA. Alotaibi1 and M. Mursaleen21 Department of Mathematics, King Abdul Aziz University, P.O.Box 80203 Jeddah, KSA2 Department of Mathematics, Aligarh Muslim University, Aligarh, IndiaReceived: May 7, 2011; Revised Nov. 21, 2011; Accepted 16 Dec. 2011Published online: 1 Sep. 2012Abstract: In this paper we study A-statistical summability of conjugate Fourier series, derived Fourier series and Walsh-Fourier series.At the end of the paper it is shown that A-statistical summability version is stronger than A-summability.Keywords: Statistical convergence; A-statistical convergence; Fourier series; Walsh-Fourier series.1. IntroductionLet K = {ki} be an index set and let ϕK be the character-istic sequence of K, i.e. ϕK = (ϕKj ) withϕKj ={1, for j ∈ K;0, otherwise.If ϕK is C1-summable then the limitlimnn∑j=1ϕKjis called the asymptotic density of K and is denoted byδ(K). A sequence x = (xk) of real numbers is said tobe statistically convergent to L if δ(Kε) = 0 for everyε > 0, where Kε := {k ∈ N : |xk − L| ≥ ε} ( cf. [4],[7]). In this case L is called the st-limit of x.Let c and l∞ denote the set of all convergent and boundedsequences, respectively. Let X and Y be two sequencespaces and A = (ank)∞n;k=1 be an infinite matrix of realor complex numbers. We write Ax = (An(x)) providedAn(x) =∑k ankxk converges for each n. A sequencex = (xk) is said to be A-summable to L if limnAn(x) =L. If x = (xk) ∈ X implies that Ax ∈ Y , then we saythat A defines a matrix transformation from X into Y andby (X,Y ) we denote the class of such matrices. If X andY are equipped with the limits X-lim and Y -lim, respec-tively, A ∈ (X,Y ) and Y -limnAn(x) = X-limk xk forall x ∈X, then we say thatA is a regular map fromX intoY and in this case we write A ∈ (X,Y )reg. The matricesA ∈ (c, c)reg are called regular.For a non-negative regular matrix A = (ank)∞n,k=1,following Freedman and Sember [6], an index set K ={ki} is said to have A-densityδA(K) = limnAnϕKif AϕK ∈ c. ThusδA(K) = limn∑k∈Kank = limn∑ian,ki .A sequence x is said to beA-statistically convergentto L if δA(Kε) = 0 for every ε > 0. In this case we writestA-limx = L, and xkstA−→ L. By the symbol stA wedenote the set of all A-statistically convergent sequences.Note that c ⊂ stA.The following characterization is due to Kolk [9, Corol-lary 4] which will be needed in our results.Lemma 11B = (bnk)∞n,k=1 ∈ (c, stA∩ l∞)reg if and onlyif(i)‖ B ‖= ∑k | bnk |< ∞, and there exists N = {ni}so that δA(N) = 1 and(ii)limi bnik = 0 (k ∈ N ), i.e. stA-lim bnk = 0 for eachk,∗ Corresponding author: e-mail: aalotaibi@kau.edu.sac© 2012 NSPNatural Sciences Publishing Cor.536 A. Alotaibi and M. Mursaleen: A-Statistical Summability of Fourier Series and Walsh-Fourier Series(iii)limi∑k bnik = 1, i.e. stA-lim∑k bnk = 1.Zygmund proved some theorems on the statistical con-vergence of Fourier series in the first edition of his book( [12], pp 181-188 ). Recently, A-statistical convergencehave been used for some approximating operators in [1],[3] and [8]; and for Fourier integrals [10]. In this paper, weprove some results on statistical summability of conjugateFourier series, derived Fourier series and Walsh-Fourierseries.2. Preliminaries2.1.Let f be L-integrable and periodic with period 2π, andlet the Fourier series of f be1a0+∞∑k=1(ak cos kx+ bk sin kx) . (2.1)Then the series conjugate to it is∞∑k=1(bk cos kx− ak sin kx) , (2.2)with partial sum S̃n(x); and the derived series is∞∑k=1k (bk cos kx− ak sin kx) , (2.3)with partial sum S′n(x). We writeψx(t) = ψ(f, t){= f(x+ t) − f(x− t) , for 0 < t ≤ π;g(x) , for t = 0 ; (2.4)andβx(t) =ψx(t)4 sin 12 t, (2.5)where g(x) = f(x+ 0) − f(x− 0).We shall need the following lemma which is statisti-cal version of the Banach Weak Convergence Theorem[2].Lemma 21stA-limn→∞∫ π0gndhx = 0 for all hx ∈BV [0, π], if and only if ‖ gn ‖< ∞ for all n and stA-limn→∞ gn = 0 ; where BV [0, π] denotes the set ofall functions of bounded variations on [0, π].Parallel version also holds if [0, π] is replaced by [0, 1].2.2.Let us define a sequence of functionsh0(x), h1(x), ......, hn(x)which satisfy the following conditions:h0(x) ={1 , for 0 ≤ x ≤ 12 ;−1 , for 12 ≤ x < 1 ;(2.6)h0(x + 1) = h0(x) and hn(x) = h0(2nx), n =1, 2, ......The functions hn(x) are called theRademacher′sfunctions.The Walsh functions are defined by φ0(x) = 1,φn(x) = hn1(x)hn2(x).......hnr (x), 0 ≤ x ≤ 1,(2.7)for n = 2n1 +2n2 + ......2nr ; where the integers ni areuniquely determined by ni+1 < ni.Let f be L-integrable and periodic with period 1, andlet the Walsh-Fourier series of f be∞∑n=1cnφn(x), (2.8)wherecn =∫ 10f(x)φn(x)dx, (2.9)are called the Walsh-Fourier coefficients of f.Let us recall some basic properties of Walsh functions(see [4]).For each fixed x ∈ [0, 1) and for all t ∈ [0, 1)(i)φn(x+̇t) = φn(x)φn(t),(ii)∫ 10f(x+̇t)dt =∫ 10f(t)dt, and(iii)∫ 10f(t)φn(x+̇t)dt =∫ 10f(x+̇t)φn(t)dt;where +̇ denotes the operation in the dyadic group,the set of all sequences s = (sn), sn = 0, 1 for n =1, 2, ....., is addition modulo 2 in each coordinate.Let for x ∈ [0, 1),Jk(x) =∫ x0φk(t)dt, k = 0, 1, 2, ... (2.10)It is easy to see that Jk(x) = 0 for x = 0, 1.3. Main resultsWe prove the following theorems:Theorem 31LetB = (bnk)∞n,k=1 ∈ (c, stA∩l∞)reg . Thenfor every x ∈ [−π, π] for which βx(t) ∈ BV [0, π],stA- limm∞∑n=1bmnS′n(x) = βx(0+) (3.1.1)if and only ifstA- limm∞∑n=1bmn sin(n+12)t = 0 for all t ∈ [0, π].(3.1.2)Theorem 32Let B ∈ (c, stA ∩ l∞)reg. Then for every x ∈[−π, π] for which βx(t) ∈ BV [0, π],stA- limm∞∑n=1nbmnS̃n(x) =1πg(x) (3.2.1)c© 2012 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci. 6, No. 3, 535-538 (2012) / www.naturalspublishing.com/Journals.asp 537if and only ifstA- limm∞∑n=1bmn cosnt = 0 for all t ∈ [0, π]. (3.2.2)Theorem 33LetB ∈ (c, stA∩l∞)reg . Let zk(x) = ckφk(x)for an L-integrable function f ∈ BV [0, 1). Then for everyx ∈ [0, 1)stA- limn∞∑k=1bnkzk(x) = 0 (3.3.1)if and only ifstA- limn∞∑k=1bnkJk(x) = 0, (3.3.2)where x is a point at which f(x) is of bounded variation.4. ProofsProof(Proof of Theorem 3.1. ). We haveS′n(x) =1π∫ π0ψx(t)(n∑k=1k sin kt)dt= − 1π∫ π0ψx(t)ddt(sin(n+ 12 )t2 sin t2)dt,= In +2π∫ π0sin(n+12)t dβx(t), (3.1.3)whereIn =1π∫ π0βx(t) cost2(sin(n+ 12 )tsin t2)dt. (3.1.4)Then∞∑n=1bmnS′n(x) =∞∑n=1bmnIn +2π∫ π0Lm(t) dβx(t),(3.1.5)whereLm(t) =∞∑n=1bmn sin(n+12)t. (3.1.6)Since βx(t) is of bounded variation on [0, π] and βx(t) →βx(0+) as t → 0, βx(t) cos t2 has also the same property.Hence by Jordan’s convergence criterion for Fourier seriesIn → βx(0+) as n→ ∞.SinceB ∈ (c, stA∩l∞)reg , by the conditions of Lemma1.1, we getstA- limm∞∑n=1bmnIn = βx(0+). (3.1.7)Now, it is enough to show that (3.1.2) holds if and only ifstA- limm∫ π0Lm(t) dβx(t) = 0. (3.1.8)Hence, by Lemma 2.1, it follows that (2.1.8) holds if andonly if‖ Lm(t) ‖≤M for all m and for all t ∈ [0, π], (3.1.9)and (3.1.2) holds. Since (3.1.9) is satisfied by Lemma 1.1(i),it follows that (3.1.8) holds if and only if (3.1.2) holds.Hence the result follows immediately.Proof(Proof of Theorem 3.2). We haveWe haveS̃n(x) =1π∫ π0ψx(t) sinnt dt,=g(x)nπ+1nπ∫ π0cosnt dψx(t).Therefore∞∑n=1nbmnS̃n(x) =g(x)π∞∑n=1bmn +1π∫ π0Km(t) dψx(t),(3.2.3)whereKm(t) =∞∑n=1bmn cosnt.Now, taking stA-lim on both sides of (3.2.3) and usingLemmas 1.1 and 2.1 as in the proof of Theorem 3.1, weget the required result.Proof(Proof of Theorem 3.3). We havezk(x) = ckφk(x) =∫ 10f(t)φk(t)φk(x)dt,=∫ 10f(t)φk(x+̇t)dt =∫ 10f(x+̇t)φk(t)dt,where x+̇t belongs to the set Ω of dyadic rationals in[0, 1), in particular each element of Ω has the form p/2nfor some non-negative integers p and n, 0 ≤ p < 2n.Now,on integration by parts, we obtainzk(x) = [f(x+̇t)Jk(t)]10 −∫ 10Jk(t)df(x+̇t),= −∫ 10Jk(t)df(x+̇t), since Jk(x) = 0 for x = 0, 1.Hence∞∑k=1bnkzk(x) = −∫ 10Dn(t) dhx(t), (3.3.3)c© 2012 NSPNatural Sciences Publishing Cor.538 A. Alotaibi and M. Mursaleen: A-Statistical Summability of Fourier Series and Walsh-Fourier SerieswhereDn(t) =∞∑k=1bnkJk(t), (3.3.4)and hx(t) = f(x+̇t).Write, for any t ∈ R, gn = (Dn(t)).Since B ∈ (c, stA ∩ l∞)reg , it follows by Lemma 1.1that ‖ gn ‖<∞ for all n, and gn stA−→ 0. Hence by Lemma2.1, ∫ 10Dn(t) dhx(t)stA−→ 0.Now, taking stA-lim in (3.3.3) and (3.3.4) and usingLemma 2.1, we get the desired result.Remark. Theorem 3.3 generalizes the result of Mursaleen[11] and the following example shows that this theorem isstronger than that of [11].Example. Let A be a (C, 1) matrix, B = I (identitymatrix). Let us define a sequence u = (uk) byuk :={1, if k = m2,m ∈ N ;0, otherwise.Then u is A-statistically convergent to 0 but not A-summable. Now write ẑk(x) = (1+uk)zk(x) and Ĵk(x) =(1 + uk)Jk(x), it is easy to see that Theorem 3.3 holds ifwe replace zk(x) by ẑk(x) and Jk(x) by Ĵk(x) but Theo-rem 3.1 of [11] does not hold.References[1] O. Agratini , A-Statistical convergence of a class of integraloperators, Applied Math. & Information Sci. (to appear).[2] S. Banach, Théorie des operations lineaires, Warszawa,1932.[3] O. Duman, M. K. Khan and C. Orhan, A-Statistical con-vergence of approximating operators, Math. Ineq. Appl.,6(4)(2003) 689-699.[4] H. Fast, Sur la convergence statistique, Colloq. Math.,2(1951) 241-244.[5] N. J. Fine, On the Walsh-functions, Trans. Amer. Math. Soc.,65(1949) 372-414.[6] A. R. Freedman and J. J. Sember, Densities and summabil-ity, Pacific J. Math., 95(1981) 293-305.[7] J. A. Fridy, On statistical convergence, Analysis, 5(1985)301-313.[8] N. İspir, A-Statistical approximation by Jayasri operators,Math. Comm., 8(2003) 85-94.[9] E. Kolk, Matrix maps into the space of statistically conver-gent bounded sequences, Proc. Estonia Math. Soc. Phys.Math., 45(1996) 187-192.[10] F. Moricz, Strong Cesàro summability and statistical limitof Fourier integrals, Analysis, 25(2005) 79-86.[11] Mursaleen, Application of infinite matrices to Walsh func-tions, Demons. Math., 27 (1994) 279-282.[12] A. Zygmund, Trigonometric Series, 2nd Ed., CambridgeUniv. Press, 1959.c© 2012 NSPNatural Sciences Publishing Cor.

English

A-Statistical Summability of Fourier Series and Walsh-Fourier Series

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=6AB1F49BBA168494FEA77EEAB1E47292?doi=10.1.1.1037.5552&rep=rep1&type=pdf

A-Statistical Summability of Fourier Series and Walsh-Fourier

Abstract

Similar works

Full text

Available Versions

CiteSeerX

Arab Journals Platform