ARCHIVUM MATHEMATICUM (BRNO)
Tomus 41 (2005), 423 – 437
INTEGRABILITY AND L1 - CONVERGENCE
OF REES-STANOJEVIĆ SUMS WITH GENERALIZED
SEMI-CONVEX COEFFICIENTS OF NON-INTEGRAL ORDERS
KULWINDER KAUR
Abstract. Integrability and L1 −convergence of modified cosine sums introduced by Rees and Stanojević under a class of generalized semi-convex null
coefficients are studied by using Cesàro means of non-integral orders.
1. Introduction
Let
(1.1)
g(x) =
∞
X
ak cos kx
k=1
be the Fourier cosine series.
The problem of L1 -convergence of the Fourier cosine series (1.1) has been settled
for various special classes of coefficients. Young [9] found that an log n = o(1),
n → ∞ is a necessary and sufficient condition for the L1 -convergence of the cosine
series with convex (△2 an ≥ 0) coefficients, and Kolmogorov [8] extended this
∞
P
result to the cosine series with quasi-convex
n △2 an < ∞ coefficients and
proved the following well known theorem:
n=1
Theorem 1.1. If {an } is a quasi-convex null sequence, then for the L1 -convergence
of the cosine series (1.1), it is necessary and sufficient that lim an log n = 0.
n→∞
Definition ([6]). A sequence {an } is said to be semi-convex, if
an → 0 as
n → ∞,
2000 Mathematics Subject Classification: 42A20, 42A32.
Key words and phrases: L1 -convergences, Cesàro means, conjugate Cesàro mean, semi-convex
null coefficients, generalized semi-convex null coefficients, Fourier cosine series.
Received March 9, 2004.
424
K. KAUR
and
∞
X
n △2 an−1 + △2 an < ∞ ,
(1.2)
(a0 = 0)
n=1
where
△2 an = △a n − △an+1 .
It may be remarked here that every quasi-convex null sequence is semi- convex.
In 1968, Kano [6] generalized Theorem 1.1 in the following form:
Theorem 1.2. If {ak } is semi-convex null sequence, then (1.1) is a Fourier series,
or equivalently it represents an integrable function.
Later on, Garrett and Stanojević [5] introduced modified cosine sums
n
n X
n
X
1X
(1.3)
gn (x) =
△ak +
(△aj ) cos kx
2
k=0
k=1 j=k
and proved the following theorem.
Theorem 1.3. Let {an } be a null sequence of bounded variation. Then the sequence of modified cosine sums
(1.4)
gn (x) = Sn (x) − an+1 Dn (x)
where Sn (x) are the partial sums of the cosine series (1.1) and Dn (x) is the Dirichlet kernel, converges in L1 -norm to g(x), the pointwise sum of the cosine series,
if and only if for every ǫ > 0, there exists δ(ǫ) > 0, independent of n, such that
Zδ X
∞
(1.5)
|△ak Dk (x)| dx < ε ,
for every
n.
0 k=n+1
This result contains as a special case a number of classical and neo-classical
results. In particular, in [5] the following corollary to Theorem 1.3 is proved.
Theorem 1.4. Let {an } be a null sequence of bounded variation satisfying condition (1.5). Then the cosine series is the Fourier series of its sum g(x) and
kSn − gk = o(1), n → ∞ is equivalent to an log n = o(1), n → ∞.
In [4] Garrett and Stanojević proved the following theorem:
Theorem 1.5. If {an } is a null quasi-convex sequence, then gn (x) converges to
g(x) in the L1 -norm.
We generalize semi-convexity of null sequence in the following way:
Definition ([7]). A sequence {an } is said to be generalized semi-convex, if
an → 0
as n → ∞
and
(1.6)
∞
X
n=1
nα (△α+1 an−1 + △α+1 an ) < ∞ ,
for α > 0
(a0 = 0) .
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
425
For α = 1, this class reduces to the class defined in [6].
In [7] Kaur, K. and Bhatia, S. S. proved the following theorem:
Theorem 1.6. If
an → 0
as
n→∞
and
∞
X
n=1
nα (△α+1 an−1 + △α+1 an ) < ∞ ,
where α > 0 is an integral (a0 = 0), then gn (x) converges to g(x) in L1 -metric if
and only if △an log n = o(1), as n → ∞.
The object of this paper is to show that the above mentioned Theorem of Kaur,
K. and Bhatia, S. S. holds good for non-integral values of α > 0.
2. Notation and formulae
In what follows, the following notations are used [10]:
Given a sequence S0 , S1 , S2 , . . . , we define for every α = 0, 1, 2, . . . , the sequence
S0α , S1α , S2α , . . . , by the conditions:
Sn0 = Sn
Snα = S0α−1 + S1α−1 + S2α−1 + · · · + Snα−1
(α = 1, 2, . . . , n = 0, 1, 2, . . . ) .
α
α
Similarly for α = 0, 1, 2, . . . , we define the sequence of numbers Aα
0 , A1 , A2 , . . . ,
by the conditions
A0n = 1 ,
α−1
Aα
+ Aα−1
+ Aα−1
+ · · · + Aα−1
(α = 1, 2, . . . , n = 0, 1, 2, . . . ) .
n = A0
n
1
2
P
Consider
an be
P a given infinite series. For any real number α the Cesàro
sums of order α of
an are defined by
Snα (ap ) = Snα =
n
X
Aα
n−p ap =
p=0
n
X
Aα−1
n−p Sp ,
p=0
where Aα
p denotes the binomial coefficients and are given by the following relations
∞
X
p
−α−1
Aα
p x = (1 − x)
∞
X
Spα xp = (1 − x)−α
p=0
and Sn ’s are given by
(2.1)
p=0
∞
X
p=0
Sp xp .
426
K. KAUR
Also
Aα
n =
n
X
Aα−1
,
p
α
α−1
Aα
n − An−1 = An
p=0
nα
n+α
α
An =
≃
(α 6= −1, −2, −3, . . . ) .
n
Γ(α + 1)
The Cesàro means Tkα of order α are then defined by
Sα
Tkα = kα .
Ak
Also for 0 < x ≤ π, let
x
1
D̄0 (x) = − cot
2
2
S̄n (x) = D̄0 (x) + sin x + sin 2x + · · · + sin nx
S̄n1 (x) = S̄0 (x) + S̄1 (x) + S̄2 (x) + · · · + S̄n (x)
S̄n2 (x) = S̄01 (x) + S̄11 (x) + S̄21 (x) + · · · + S̄n1 (x)
..
.
S̄nk (x) = S̄0k−1 (x) + S̄1k−1 (x) + S̄2k−1 (x) + · · · + S̄nk−1 (x) .
The conjugate Cesàro means T̄kα of order α are defined by
T̄kα =
(2.2)
S̄kα
.
Aα
k
The following formulae will also be needed:
S̄nα (S̄pr ) = S̄nα+r+1 ,
(2.3)
α+1
S̄nα+1 − S̄n−1
= S̄nα ,
(2.4)
n
X
β
α+β+1
Aα
.
n−p Ap = An
p=0
For any positive integer α the differences of order α of the sequence {an } are
defined by the equations
△1 an = an − an+1 ,
△α an = △(△α−1 an ) ,
n = 0, 1, 2, . . . .
For these differences we have
α
∞
X
X
−α−1
−α−1
(2.5)
△ α an =
Am
an+m =
Am
an+m ,
m=0
−α−1
Am
m=0
since
= 0 for m > α + 1.
If the series (2.5) are convergent for some α which is not a positive integer, then
we denote the differences
∞
X
−α−1
(2.6)
△ α an =
Am
an+m ,
n = 0, 1, 2, . . . .
m=0
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
427
The broken differences △α
n ap are defined by
△α
n ap
(2.7)
=
n−p
X
−α−1
Am
ap+m .
m=0
By repeated partial summation of order α,
n
n
X
X
S̄pα−1 (ap )△α
(2.8)
ap b p =
n bp .
p=0
p=0
If α is positive integer then we have
n
X
(2.9)
ap b p =
p=0
n−α
X
S̄pα−1 (ap )△α bp +
n
X
S̄pα−1 (ap )△α
n bp .
p=n−α+1
p=0
3. Lemmas
We need the following lemmas for the proof of our result:
Lemma 3.1 ([3]). If α > 0, p > 0,
(i)
ǫn = o(n−p ),
∞
P
(ii)
Anα+p △α+1 ǫn < ∞,
n=0
then
(iii)
∞
P
λ+1 △
Aλ+p
ǫn < ∞, for −1 ≤ λ ≤ α and
n
n=0
Aλ+p
△λ ǫ n
n
(iv)
n → ∞.
is of bounded variation for 0 ≤ λ ≤ α and tends to zero as
Lemma 3.2 ([1]). Let r be the real number > 0. If the sequence {ǫn } satisfies the
conditions:
(i)
ǫn = O(1) and
∞
P
(ii)
nr △r+1 ǫn < ∞,
n=1
then
β
△ ǫn =
∞
X
r+1
Ar−β
ǫn+m ,
m △
for
β > 0.
m=0
Lemma 3.3 ([2]). If 0 6 δ ≤ 1 and 0 ≤ m < n,
m
X
δ−1 An−i
Si ≤ max
i=0
then
0≤p≤m
δ
S .
p
Lemma 3.4 ([10]). Let S̄n (x) and T̄nα be the n-th partial sum and conjugate Cesàro
mean of order α > 0, respectively, of the series
D̄0 (x) + sin x + sin 2x + sin 3x + · · · + sin nx + . . .
Then
(i)
Rπ S̄n (x) dx ∼ log n,
0
428
(ii)
K. KAUR
Rπ α T̄n dx remains bounded for all n.
0
4. Main result
The main result of this paper is the following theorem:
Theorem 4.1. If
an → 0
n→∞
as
and
∞
X
n=1
nα (△α+1 an−1 + △α+1 an ) < ∞ ,
where α > 0 is non-integral (a0 = 0), then gn (x) converges to g(x) in L1 -metric if
and only if △an log n = o(1), as n → ∞.
Proof. We have
n
(4.1)
gn (x) =
n
k=0
=
=
=
n
XX
1X
△ak +
(△aj ) cos kx
2
a0
+
2
n
X
k=1 j=k
n
X
ak cos kx − an+1 Dn (x)
k=1
ak cos kx − an+1 Dn (x)
k=1
n−1
X
(ak−1 − ak+1 )
k=1
+ an
(a0 = 0)
sin nx
sin kx
+ an−1
2 sin x
2 sin x
sin(n + 1)x
− an+1 Dn (x) ,
2 sin x
where
Dn (x) =
gn (x) =
sin nx + sin(n + 1)x
,
2 sin x
n−1
X
(ak−1 − ak+1 )
k=1
sin kx
sin nx
sin(n + 1)x
+ an−1
+ an
2 sin x
2 sin x
2 sin x
sin(n + 1)x
sin nx
− an+1
2 sin x
2 sin x
n
1 X
sin(n + 1)x
=
.
(△ak−1 + △ak ) sin kx + △an
2 sin x
2 sin x
− an+1
k=1
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
429
Applying Abel’s transformation, we have
gn (x) =
n−1
k
i
X
1 hX 2
(△ ak−1 + △2 ak )
sin vx
2 sin x
v=1
k=1
n
i
X
sin(n + 1)x
1 h
(△an−1 + △an )
sin vx + △an
+
2 sin x
2 sin x
v=1
=
n−1
i
1 hX 2
(△ ak−1 + △2 ak )(S̄k0 (x) − S̄0 (x))
2 sin x
k=1
sin(n + 1)x
1 (△an−1 + △an ) S̄n0 (x) − S̄0 (x) + △an
+
2 sin x
2 sin x
n−1
n−1
h
i
X
X
1
=
(△2 ak−1 + △2 ak )S̄k0 (x) −
(△2 ak−1 + △2 ak )S̄0 (x)
2 sin x
k=1
(4.2)
k=1
1 (△an−1 + △an )S̄n0 (x) − (△an−1 + △an )S̄0 (x)
+
2 sin x
sin(n + 1)x
+ △an
2 sin x
n−1
h
i
X
1
(△2 ak−1 + △2 ak )S̄k0 (x) + (△an−1 + △an )S̄n0 (x) + a2 S̄0 (x)
=
2 sin x
k=1
sin(n + 1)x
.
+ △an
2 sin x
As α > 0 is non-integral. Let α = r + δ, r is the integral part of α, and δ is the
fractional part, 0 < δ < 1.
Case (i). Let r = 0.
Applying Abel’s transformation of order −δ + 1, we have by (2.8)
n−1
X
S̄kδ−1 (x) △δ+1 ak−1 + △δ+1 ak
k=1
=
n−1
X
n−(k+1)
X
S̄k (x)
δ−2
Am
(△δ+1 am+k−1 + △δ+1 am+k )
m=1
k=1
Also by Lemma 3.2, we have
n−1
X
k=1
n
X
n−1
S̄kδ−1 (x) △δ+1 ak−1 + △δ+1 ak =
S̄k (x) △2 ak−1 + △2 ak
k=1
−
∞
X
m=n−k
=
n−1
X
k=1
o
δ−2
Am
(△δ+1 am+k−1 + △δ+1 am+k )
S̄k (x) △2 ak−1 + △2 ak − Rn (x)
430
K. KAUR
where
Rn (x) =
n−1
X
δ−2
S̄k (x){An−k
(△δ+1 an−1 + △δ+1 an )
k=1
δ−2
+ An−k+1
(△δ+1 an + △δ+1 an+1 ) + . . . } .
Therefore
n−1
1 X
S̄k (x) △2 ak−1 + △2 ak
2 sin x
k=1
=
n−1
o
1 n X δ−1
S̄k (x) △δ+1 ak−1 + △δ+1 ak + Rn (x)
2 sin x
k=1
and consequently by (4.2)
n−1
gn (x) =
1 n X δ−1
S̄k (x) △δ+1 ak−1 + △δ+1 ak
2 sin x
k=1
o
sin(n + 1)x
+ Rn (x) + (△an−1 + △an )S̄n0 (x) + a2 S̄0 (x) + △an
.
2 sin x
When r = 0, then α = δ and
g(x) = lim gn (x)
n→∞
=
∞
o
1 n X δ−1
S̄k (x) △δ+1 ak−1 + △δ+1 ak + a2 S̄0 (x)
2 sin x
k=1
Therefore,
Z
0
π
π
∞
1 n X δ−1
S̄k (x)(△δ+1 ak−1 + △δ+1 ak )
2
sin
x
0
k=n
Z π
o
sin(n + 1)x − Rn (x) − (△an−1 + △an )S̄n0 (x) dx +
dx
△an
2 sin x
0
Z
∞
nX
δ+1
π δ−1 δ+1
S̄
6C
(△ ak−1 + △ ak )
(x) dx
k
g(x) − gn (x) dx ≤
Z
0
k=n
+ |(△an−1 + △an )|
Zπ
0
+C
Z
π
|Rn (x)| dx +
0
o
0
S̄n (x)dx
Z
0
π
sin(n + 1)x dx
△an
2 sin x
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
=C
∞
nX
k=n
Akδ−1 (△δ+1 ak−1 + △δ+1 ak )
+ |(△an−1 + △an )|
Zπ
0
+C
6 C1
Zπ
0
∞
X
k=n
Zπ
+C
|Rn (x)| dx +
Z
π
0
δ−1 T̄
(x) dx
k
0
o
S̄ (x)dx
n
π
0
sin(n + 1)x dx
△an
2 sin x
Akδ−1 (△δ+1 ak−1 + △δ+1 ak ) + |(△an−1 + △an )|
|Rn (x)| dx +
0
Z
π
0
sin(n + 1)x △an
dx
2 sin x
= o(1) + C1 |(an−1 − an+1 )| + C
(4.3)
Z
Z π
Zπ
sin(n + 1)x |Rn (x)| dx + △an
dx ,
2 sin x
0
0
by Lemma 3.1 and 3.4.
Now for the estimate of
Rπ
|Rn (x)| dx, we have
0
Zπ
0
431
Zπ n−1
X
δ−2
|
S̄k (x)An−k
(△δ+1 an−1 + △δ+1 an )
|Rn (x)| dx =
k=1
0
n−1
X
+
k=1
6
π
Z
X
δ+1
n−1
(△ an−1 + △δ+1 an ) S̄k (x)Aδ−2 dx
n−k
0
+
k=1
Z
π
0
6
Z
Z
X
δ+1
n−1
(△ an + △δ+1 an+1 ) S̄k (x)Aδ−2
k=1
π
0
+
δ−2
S̄k (x)An−k+1
(△δ+1 an + △δ+1 an+1 ) + · · · | dx
δ+1
(△ an−1 + △δ+1 an )
0
max
16p6n−1
π
n−k+1 dx
δ−1 S̄
(x) dx
p
+ ...
δ+1
(△ an + △δ+1 an+1 ) max S̄pδ−1 (x) dx + . . . ,
by Lemma 3.3,
16p6n
432
K. KAUR
= △δ+1 an−1 + △δ+1 an ) Anδ−1
Zπ
max
16p6n−1
0
+ (△δ+1 an + △δ+1 an+1 ) Aδ−1
n+1
Zπ
0
δ−1 T̄
(x) dx
p
max T̄pδ−1 (x) dx + . . .
16p6n
≤ CAnδ−1 (△δ+1 an−1 + △δ+1 an )
+ CAδ−1 (△δ+1 an + △δ+1 an+1 ) + . . .
n+1
≤ CAδn (△δ+1 an−1 + △δ+1 an )
+ CAδn+1 (△δ+1 an + △δ+1 an+1 ) + . . .
= o(1) + o(1) + . . .
= o(1) ,
by Lemma 3.1 and 3.4.
Moreover, since
Z
π
0
Therefore,
sin(n + 1)x dx ≤ C log n ,
2 sin x
Z
π
0
Thus, by (4.3),
(4.4)
lim
n→∞
Zπ
n ≥ 2.
sin(n + 1)x dx ∼ △an log n .
△an
2 sin x
|g(x) − gn (x)| dx = o(1) ,
if and only if
0
△an log n = o(1) ,
as n → ∞.
Case (ii). Let r > 1.
We have
n
gn (x) =
sin(n + 1)x
1 X
.
(△ak−1 + △ak ) sin kx + △an
2 sin x
2 sin x
k=1
Applying Abel’s transformation of order r,
n−r
(4.5)
gn (x) =
1 nX
△r+1 ak−1 + △r+1 ak S̄kr−1 (x)
2 sin x
k=1
+
r
X
k=1
+ △an
o
k−1
(x) + a2 S̄0 (x)
△k an−k + △k an−k+1 S̄n−k+1
sin(n + 1)x
.
2 sin x
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
433
Again applying Abel’s transformation of order -δ, we obtain
n−1
X
S̄kα−1 (x)(△α+1 ak−1 + △α+1 ak )
k=1
=
n−1
X
n−(k+1)
S̄kr−1 (x)
X
δ−1
Am
(△α+1 am+k−1 + △α+1 am+k ) .
m=0
k=1
By Lemma 3.2,
(4.6)
n−1
1 X α−1
S̄k (x)(△α+1 ak−1 + △α+1 ak )
2 sin x
k=1
=
n−1
o
1 n X r−1
S̄k (x)(△r+1 ak−1 + △r+1 ak ) − Rn (x) ,
2 sin x
k=1
where
n−1
X
Rn (x) =
δ−1
S̄kr−1 (x){An−k
(△α+1 an−1 + △α+1 an )
k=1
δ−1
+ An−k+1
(△α+1 an + △α+1 an+1 ) + . . . }
n−1
X
=
+
δ−1
S̄kr−1 (x)An−k
(△α+1 an−1 + △α+1 an )
k=1
n−1
X
k=1
δ−1
S̄kr−1 (x)An−k+1
(△α+1 an + △α+1 an+1 ) + . . .
Replacing n by n − r + 1 in (4.6), we have
(4.7)
n−r
1 X α−1
S̄k (x)(△α+1 ak−1 + △α+1 ak )
2 sin x
k=1
=
n−r
o
1 n X r−1
S̄k (x)(△r+1 ak−1 + △r+1 ak ) − Rn−r+1 (x) .
2 sin x
k=1
Now by (4.5) and (4.7), we have
n−r
(4.8)
gn (x) =
1 n X α−1
S̄k (x)(△α+1 ak−1 + △α+1 ak ) + Rn−r+1 (x)
2 sin x
k=1
+
r
X
k=1
o
k−1
(△k an−k + △k an−k+1 )S̄n−k+1
(x) + a2 S̄0 (x)
+ △an
sin(n + 1)x
.
2 sin x
434
K. KAUR
Therefore
Z
0
π
Zπ |g(x) − gn (x)| dx ≤ 0
− Rn−r+1 (x) −
1 n
2 sin x
r
X
k=1
∞
X
S̄kα−1 (x)(△α+1 ak−1 + △α+1 ak )
k=n−r+1
o
k−1
(△k an−k + △k an−k+1 )S̄n−k+1
(x) dx
Zπ sin(n + 1)x dx
+ △an
2 sin x 0
π
Z
∞
X
α+1
(△
≤C
ak−1 + △α+1 ak ) S̄kα−1 (x) dx
k=n−r+1
+
Z
0
π
|Rn−r+1 (x)| dx
0
π
Z
r
X
k
k−1
k
S̄
+
(△ an−k + △ an−k+1 )
n−k+1 (x) dx
k=1
0
Zπ
sin(n + 1)x + △an
dx
2 sin x
0
∞
X
=C
Zπ
α+1
α−1 α+1
T̄
(△
(x) dx
Aα−1
a
+
△
a
)
k−1
k
k
k
k=n−r+1
0
Zπ
+ |Rn−r+1 (x)| dx
0
+
r
X
k−1
An−k+1
k=1
k
(△ an−k + △k an−k+1 )
Zπ sin(n + 1)x + △an
dx
2 sin x
Zπ
0
k−1
T̄
dx
n−k+1 (x)
0
≤
∞
X
C1
Aα
k
k=n−r+1
+ C1
r
X
k=1
α+1
(△
ak−1 + △α+1 ak ) + C
k−1
An−k+1
Zπ
k
(△ an−k + △k an−k+1 ) +
|Rn−r+1 (x)| dx
0
Zπ △an sin(n + 1)x dx
2 sin x 0
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
= o(1) + o(1) + C
Zπ
0
= o(1) + C
(4.9)
Zπ
0
435
Zπ sin(n + 1)x dx
|Rn−r+1 (x)| dx + △an
2 sin x 0
Zπ sin(n + 1)x |Rn−r+1 (x)| dx + △an
dx
2 sin x 0
by the hypothesis of the theorem and Lemma 3.1.
But
Zπ
0
Zπ n−r
X r−1
δ−1
|Rn−r+1 (x)| dx ≤
S̄k (x)An−r−k+1
(△α+1 an−r + △α+1 an−r+1 ) dx
+
+
0
Zπ
0
..
.
≤
k=1
0
Zπ
n−r
X
n−r
X r−1
δ−1
S̄k (x)An−r−k+2
(△α+1 an−r+1 + △α+1 an−r+2 ) dx
k=1
n−r
X r−1
δ−1
S̄k (x)An−r−k+3
(△α+1 an−r+2 + △α+1 an−r+3 ) dx
k=1
δ−1
An−r−k+1
Ar−1
k
k=1
+
Zπ
0
n−r
X
δ−1
An−r−k+2
Ar−1
k
n−r
X
δ−1
An−r−k+3
Ar−1
k
k=1
+
k=1
n−r
X
+ C1
k=1
n−r
X
k=1
..
.
0
Zπ
r−1 α+1
T̄
(x) dx (△
an−r+1 + △α+1 an−r+2 )
k
r−1 α+1
T̄
(x) dx (△
an−r+2 + △α+1 an−r+3 )
k
α+1
δ−1
(△
An−r−k+1
Ar−1
an−r + △α+1 an−r+1 )
k
k=1
n−r
X
+ C1
Zπ
0
..
.
≤ C1
r−1 α+1
T̄
(x) dx (△
an−r + △α+1 an−r+1 )
k
α+1
δ−1
(△
An−r−k+2
Ar−1
an−r+1 + △α+1 an−r+2 )
k
α+1
δ−1
△
An−r−k+3
Ar−1
an−r+2 + △α+1 an−r+3 )
k
436
K. KAUR
≤ C1
n+1−r
X
k=1
α+1
δ−1
(△
An+1−r−k
Ar−1
an−r + △α+1 an−r+1 )
k
+ C1
n+2−r
X
+ C1
n+3−r
X
k=1
k=1
..
.
α+1
δ−1
(△
An+2−r−k
Ar−1
an−r+1 + △α+1 an−r+2 )
k
α+1
δ−1
(△
An+3−r−k
Ar−1
an−r+2 + △α+1 an−r+3 )
k
α+1
r+δ−1 (△
an−r + △α+1 an−r+1 )
≤ C2 An+1−r
α+1
r+δ−1 (△
an−r+1 + △α+1 an−r+2 )
+ C2 An+2−r
+ C2 Ar+δ−1 (△α+1 an−r+2 + △α+1 an−r+3 )
n+3−r
..
.
α+1
r+δ
(△
an−r + △α+1 an−r+1 )
≤ C2 An+1−r
α+1
r+δ
(△
an−r+1 + △α+1 an−r+2 )
+ C2 An+2−r
α+1
r+δ
(△
an−r+2 + △α+1 an−r+3 )
+ C2 An+3−r
..
.
α+1
= C2 Aα
an−r + △α+1 an−r+1 )
n+1−r (△
α+1
+ C2 Aα
an−r+1 + △α+1 an−r+2 )
n+2−r (△
α+1
+ C2 Aα
an−r+2 + △α+1 an−r+4 )
n+3−r (△
..
.
= o(1) + o(1)
..
.
= o(1) ,
by the hypothesis of the theorem.
Hence (4.9) implies
(4.10)
lim
n→∞
Zπ
|g(x) − gn (x)| dx = o(1) ,
0
if and only if △an log n = o(1), as n→ ∞.
Thus by (4.4) and (4.10)
(4.11)
lim
n→∞
Zπ
0
|g(x) − gn (x)| dx = o(1) ,
INTEGRABILITY AND L1 - CONVERGENCE OF REES-STANOJEVIĆ SUMS. . .
if and only if △an log n = o(1), as n→ ∞, where α is non-integral.
437
References
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convergence theorem of Dedekind, Proc. London Math. Soc. 8 (1958), 1–52.
[2] Bosanquet, L. S., Note on the Bohr-Hardy theorem, J. London Math. Soc. 17 (1942), 166–
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[3] Bosanquet, L. S., Note on convergence and summability factors (III), Proc. London Math.
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[6] Kano, T., Coefficients of some trigonometric series, J. Fac. Sci. Shihshu University 3 (1968),
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[7] Kaur, K. and Bhatia, S. S., Integrability and L-convergence of Rees-Stanojević sums with
generalized semi-convex coefficients, Int. J. Math. Math. Sci. 30(11) (2002), 645–650.
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Lebesque, Bull. Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1923) 83–86.
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Department of Applied Sciences
GZS College of Engineering and Technology
Bathinda, Punjab – 151005, INDIA
E-mail: mathkaur@yahoo.co.in