IJMMS 30:11 (2002) 645–650
PII. S0161171202012942
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
INTEGRABILITY AND L1 -CONVERGENCE OF REES-STANOJEVIĆ
SUMS WITH GENERALIZED SEMICONVEX COEFFICIENTS
KULWINDER KAUR and S. S. BHATIA
Received 24 April 2001 and in revised form 25 October 2001
Integrability and L1 -convergence of modified cosine sums introduced by Rees and
Stanojević (1973) under a class of generalized semiconvex null coefficients are studied,
using Cesaro means of integral order.
2000 Mathematics Subject Classification: 42C10, 40G05.
1. Introduction. Let
g(x) =
gn (x) =
∞
a0 an cos nx,
+
2 n=1
(1.1)
n
n
n 1
∆ak +
∆aj cos kx.
2 k=0
k=1 j=k
(1.2)
The problem of L1 -convergence of the Fourier cosine series (1.1) has been settled
for various special classes of coefficients. Young [6] found that an log n = o(1), n → ∞
is a necessary and sufficient condition for cosine series with convex (∆2 an ≥ 0) coefficients, and Kolmogorov [5] extended this result to the cosine series with quasi-convex
∞
( n=1 n|∆2 an−1 | < ∞) coefficients. Later, Garrett and Stanojević [3] using modified
cosine sums (1.2), proved the following theorem.
Theorem 1.1. Let {an } be a null sequence of bounded variation. Then the sequence
of modified cosine sums
gn (x) = Sn (x) − an+1 Dn (x),
(1.3)
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 > 0, there exists δ() > 0, independent of n, such that
δ
∞
∆ak Dk (x)dx < ,
0
for every n.
(1.4)
k=n+1
This result contains as a special case a number of classical and neo-classical results.
In particular, in [3] the following corollary to Theorem 1.1 is proved.
646
K. KAUR AND S. S. BHATIA
Theorem 1.2. Let {an } be a null sequence of bounded variation satisfying condition
(1.4). Then the cosine series is the Fourier series of its sum g(x) and Sn (g)−g = o(1),
n → ∞ is equivalent to an log n = o(1), n → ∞.
In [2] Garrett and Stanojević proved the following theorem.
Theorem 1.3. If {an } is a null quasi-convex sequence, then gn (x) converges to
g(x) in the L1 -norm.
Definition 1.4 (see [4]). A sequence {an } is said to be semiconvex if {an } → 0 as
n → ∞, and
∞
n∆2 an−1 + ∆2 an < ∞,
(1.5)
a0 = 0 ,
n=1
where ∆ an = ∆an − ∆an+1 , ∆an = an − an+1 .
2
It may be remarked here that every quasi-convex null sequence is semi-convex. We
generalize semiconvexity of null sequences in the following way: a null sequence {an }
is said to be generalized semiconvex, if
∞
nα ∆α+1 an−1 + ∆α+1 an < ∞,
for α > 0 a0 = 0 .
(1.6)
n=1
For α = 1, this class reduces to the class defined in [4]. The object of this paper is
to show that Theorem 1.3 of Garrett and Stanojević [2] holds good for cosine sums
(1.2) with generalized semi-convex null coefficients.
2. Notation and formulae. In what follows, we use the following notions [7]:
Sn0 = Sn = a0 + a1 + · · · + an ;
Snk = S0k−1 + S1k−1 + · · · + Snk−1 ,
A0n = 1,
k = 1, 2, . . . , n = 0, 1, 2, . . . ;
Akn = Ak−1
+ Ak−1
+ · · · + Ak−1
n
0
1
(2.1)
k = 1, 2, . . . , n = 0, 1, 2, . . . .
The An ’s are called the binomial coefficients and are given by the following relation:
∞
k
(−α−1)
Aα
,
k x = (1 − x)
(2.2)
k=0
whereas Sn ’s are given by
∞
Skα x k = (1 − x)−α
k=0
∞
Sk x k ,
(2.3)
k=0
and
Aα
n=
n
Aα−1
,
v
v=0
Aα
n=
n+α
n
α
α−1
Aα
n − An−1 = An ,
(2.4)
nα
Γ (α + 1)
(α ≠ −1, −2, . . .).
The Cesaro means Tkα of order α is denoted by Tkα = Skα /Aα
k.
INTEGRABILITY AND L1 -CONVERGENCE OF REES-STANOJEVIĆ SUMS . . .
647
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),
(2.5)
..
.
S̄nk (x) = S̄0k−1 (x) + S̄1k−1 (x) + S̄2k−1 (x) + · · · + S̄nk−1 (x).
The conjugate Cesaro means T̄kα of order α is denoted by T̄kα = S̄kα /Aα
k.
We use the following lemma for the proof of our result.
Lemma 2.1 (see [1]). If α ≥ 0, p ≥ 0,
n = o n−p ,
∞
α+p An
∆α+1 n < ∞,
(2.6)
n=0
then
∞
λ+p An
∆λ+1 n < ∞
for − 1 ≤ λ ≤ α,
n=0
λ+p
An ∆λ n
(2.7)
is of bounded variation for 0 ≤ λ ≤ α and tends to zero as n → ∞.
3. Main result. The main result of this paper is the following theorem.
Theorem 3.1. If {an } is a generalized semiconvex null sequence, then gn (x) converges to g(x) in L1 -metric if and only if limn→∞ ∆an log n = o(1), as n → ∞.
Proof. We have
gn (x) =
n
n
n 1
∆ak +
∆aj cos kx
2 k=0
k=1 j=k
a0 ak cos kx − an+1 Dn (x)
+
2 k=1
n
=
=
n
ak cos kx − an+1 Dn (x)
a0 = 0
k=1
=
n−1
sin kx
sin nx
sin(n + 1)x
+ an−1
+ an
− an+1 Dn (x),
ak−1 − ak+1
2
sin
x
2
sin
x
2 sin x
k=1
(3.1)
648
K. KAUR AND S. S. BHATIA
where
sin nx + sin(n + 1)x
,
2 sin x
Dn (x) =
gn (x) =
n−1
sin kx
sin nx
sin(n + 1)x
+ an−1
+ an
ak−1 − ak+1
2
sin
x
2
sin
x
2 sin x
k=1
sin nx
sin(n + 1)x
− an+1
− an+1
2 sin x
2 sin x
(3.2)
n
1
sin(n + 1)x
∆ak−1 + ∆ak sin kx + ∆an
.
2 sin x k=1
2 sin x
=
Applying Abel’s transformation, we have
gn (x) =
n−1
k
n
1
∆2 ak−1 + ∆2 ak
sin vx + ∆an−1 + ∆an
sin vx
2 sin x k=1
v=1
v=1
sin(n + 1)x
2 sin x
n−1
0
0
1
2
2
=
∆ ak−1 +∆ ak S̄k (x) − S̄0 (x) + ∆an−1 + ∆an S̄n (x) − S̄0 (x)
2 sin x k=1
+ ∆an
sin(n + 1)x
2 sin x
n−1
n−1
1
=
∆2 ak−1 + ∆2 ak S̄k0 (x) −
∆2 ak−1 + ∆2 ak S̄0 (x)
2 sin x k=1
k=1
+ ∆an
sin(n + 1)x
1 ∆an−1 + ∆an S̄n0 (x) − ∆an−1 + ∆an S̄0 (x) + ∆an
2 sin x
2 sin x
n−1
1
∆2 ak−1 + ∆2 ak S̄k0 (x) − ∆an−1 + ∆an S̄n0 (x) + a2 S̄0 (x)
=
2 sin x k=1
+
+ ∆an
sin(n + 1)x
.
2 sin x
(3.3)
If we use Abel’s transformation α times, we have similarly,
n−α
α
α−1
1
k−1
α+1
α+1
k
∆
ak−1 + ∆
ak S̄k (x) +
∆ an−k S̄n−k+1 (x)
gn (x) =
2 sin x k=1
k=1
1
+
2 sin x
α
(3.4)
sin(n + 1)x
k−1
.
∆k an−k+1 S̄n−k+1
(x) + a2 S̄0 (x) + ∆an
2 sin x
k=1
Since S̄n (x) and T̄n (x) are uniformly bounded on every segment [, π − ], > 0.
g(x) = lim gn (x)
n→∞
=
1
2 sin x
∞
α+1
ak−1 + ∆α+1 ak S̄kα−1 (x) + a2 S̄0 (x) .
∆
k=1
(3.5)
INTEGRABILITY AND L1 -CONVERGENCE OF REES-STANOJEVIĆ SUMS . . .
649
Thus
g(x)−gn (x)=
1
2 sin x
−
α
α+1
k−1
ak−1+∆α+1 ak S̄kα−1 (x)−
∆k an−k S̄n−k+1
(x)
∆
∞
k=n−α+1
1
2 sin x
π
g(x) − gn (x) ≤ C
0 α
k=1
k−1
∆k an−k+1 S̄n−k+1
(x) − ∆an
k=1
∞
k=n−α+1
sin(n + 1)x
,
2 sin x
α+1
ak−1 + ∆α+1 ak S̄kα−1 (x)dx
∆
α
π α
π k−1
k−1
k
k
∆ an−k S̄n−k+1 (x)dx+
∆ an−k+1 S̄n−k+1 (x)dx
+C
0
0
k=1
k=1
π ∆an sin(n + 1)x dx,
+
2 sin x 0
g(x) − gn (x) ≤ C
∞
α+1
∆
ak−1 + ∆α+1 ak π
k=n−α+1
+C
α−1
S̄
(x)dx
k
0
α
α
k
π
π
∆ an−k S̄ k−1 (x)dx+ ∆k an−k+1 S̄ k−1 (x)dx
n−k+1
n−k+1
0
k=1
0
k=1
π sin(n + 1)x +
∆an 2 sin x dx
0
∞
≤C
α+1
∆
Aα
ak−1 + ∆α+1 ak k
k=n−α+1
+C
α
k=1
+C
α
Akn−k+1 ∆k an−k π
k=1
0
α
T̄ (x)dx
k
k
T̄
n−k+1 (x) dx
0
Akn−k+1 ∆k an−k+1 π
π
0
k
T̄
n−k+1 (x) dx
π ∆an sin(n + 1)x dx.
+
2 sin x 0
(3.6)
The first three terms of the above inequality are of o(1) by Lemma 2.1 and the
hypothesis of Theorem 3.1.
Moreover, since
π sin(n + 1)x 2 sin x dx ≤ C log n,
0
n ≥ 2,
(3.7)
therefore
π ∆an sin(n + 1)x dx ∼ ∆an log n.
2 sin x 0
(3.8)
650
K. KAUR AND S. S. BHATIA
π
It follows that 0 |g(x)−gn (x)|dx → 0, if and only if ∆an log n → o(1) as n → ∞. This
completes the proof of the theorem.
Acknowledgment. The authors are thankful to the referees for their wise comments, which have definitely improved the representation of the paper.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
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J. W. Garrett and Č. V. Stanojević, On integrability and L1 convergence of certain cosine
sums, Notices Amer. Math. Soc. (1975), no. 22(A-166).
, On L1 convergence of certain cosine sums, Proc. Amer. Math. Soc. 54 (1976), 101–
105.
T. Kano, Coefficients of some trigonometric series, J. Fac. Sci. Shinshu Univ. 3 (1968), 153–
162.
A. N. Kolmogorov, Sur l’ordre de grandeur des coefficients de la series de Fourier-Lebesgue,
Bull. Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1923), 83–86.
W. H. Young, On Fourier series of bounded function, Proc. London Math. Soc. 12 (1913),
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Kulwinder Kaur: School of Basic and Applied Sciences, Thapar Institute of Engineering and Technology, P.O. Box 32, Patiala 147004, India
E-mail address: mathkk@hotmail.com
S. S. Bhatia: School of Basic and Applied Sciences, Thapar Institute of Engineering
and Technology, P.O. Box 32, Patiala 147004, India
E-mail address: ssbhatia63@yahoo.com