MATEMATIQKI VESNIK
UDK 517.521
originalni nauqni rad
research paper
61, 3 (2009), 219–226
September 2009
ON L1 -CONVERGENCE OF CERTAIN GENERALIZED
MODIFIED TRIGONOMETRIC SUMS
Karanvir Singh and Kulwinder Kaur
Abstract. In this paper we define new modified generalized sine sums Knr (x) =
n
P
1
(4r ak−1 − 4r ak+1 )S̃kr−1 (x) and study their L1 -convergence under a newly defined
2 sin x k=1
class Kα . Our results generalize the corresponding results of Kaur, Bhatia and Ram [6] and
Kaur [7].
1. Introduction
Let
K(x) =
∞
P
ak cos kx
(1.1)
k=1
and
Kn (x) =
n P
n
P
1
(4aj−1 − 4aj+1 ) sin kx.
2 sin x k=1 j=k
(1.2)
Using modified cosine sums
gn (x) =
n
n P
n
P
1 P
4ak +
(4aj ) cos kx
2 k=0
k=1 j=k
of Garett and Stanojević [4], Kaur and Bhatia [5] proved the following theorem
under the class of generalized semi-convex coefficients.
Theorem 1. If {an } is a generalized semi-convex null sequence, then gn (x)
converges to K(x) in the L1 -metric if and only if 4an log n = o(1), as n → ∞.
The above mentioned result motivated the authors [6] to define a new modified
sums (1.2) and to study these sums under a different class K of coefficients in the
following way.
AMS Subject Classification: 42A20, 42A32.
Keywords and phrases: L1 −convergence; conjugate Cesàro means; generalized sine sums.
219
220
K. Singh, K. Kaur
Theorem 2. Let k be a positive real number. If
ak = o(1),
and
∞
P
k→∞
(1.3)
k|42 ak−1 − 42 ak+1 | < ∞,
(1.4)
k=1
then Kn (x) converges to K(x) in the L1 -norm.
Any sequence satisfying (1.3) and (1.4) is said to belong to the class K [6].
In particular in [6] the following corollary to Theorem 2 is proved.
Theorem 3. If {an } belongs to the class K, then the necessary and sufficient
condition for L1 -convergence of the cosine series (1.1) is limn→∞ an log n = 0.
Definition. Let α be a positive real number. If (1.3) holds and
∞
P
k α |4α+1 ak−1 − 4α+1 ak+1 | < ∞,
(a0 = 0),
k=1
then we say that {an } belongs to the class Kα .
For α = 1, the class Kα reduces to the class K.
Applying Abel’s transformation to Kn (x), we can easily see that
Kn (x) =
n
P
1
(4ak−1 − 4ak+1 )S̃k0 (x),
2 sin x k=1
where S̃k0 (x) = D̃k (x) = sin x + sin 2x + sin 3x + · · · + sin nx. So in [7] the author
n
P
1
studied the L1 -convergence of Kn (x) =
(4ak−1 − 4ak+1 )S̃k0 (x) to K(x)
2 sin x k=1
by proving the following result.
Theorem 4. Let the sequence {ak } belong to class Kα . Then Kn (x) converges
to K(x) in the L1 -norm.
If we take α = 1, then this theorem reduces to Theorem 2.
It is natural to seek ways to prove the L1 -convergence of the generalized sums
of the form
n
P
1
(4r ak−1 − 4r ak+1 )S̃kr−1 (x),
(1.5)
Knr (x) =
2 sin x k=1
where r is any real number greater than or equal to 1. It is obvious that for r = 1,
Knr (x) reduces to Kn (x).
The purpose of this paper is to prove the L1 -convergence of Knr (x) to K(x)
under the class Kα .
On L1 -convergence of certain generalized modified trigonometric sums
221
2. Notation and Formulae
We use the following notations [10].
Given a sequence S0 , S1 , S2 , . . . , we define the sequence S0α , S1α , S2α , . . . for
every α = 0, 1, 2, . . . by the conditions Sn0 = Sn , Snα = S0α +S1α−1 +S2α−1 +· · ·+Snα−1
(α = 1, 2, . . . ; n = 0, 1, 2, . . . ). Similarly we define the sequence of numbers Aα
0,
α−1
α−1
α
0
α
Aα
A
=
A
+
A
+
n
1 , A2 , . . . for α = 0, 1, 2, . . . by the conditions An = 1,
0
1
P
α−1
Aα−1
+
·
·
·
+
A
(α
=
1,
2,
.
.
.
;
n
=
0,
1,
2,
.
.
.
).
Let
a
be
a
given
infinite
n
n
2
P
series. The conjugate Cesàro sums of order α of
an for any real number α are
defined by
n
n
P
P
S̃nα (ap ) = S̃nα =
Aα
Aα−1
n−p ap =
n−p S̃p ,
p=0
p=0
where S̃n = S̃n0 = D̃n , and Aα
p denotes the binomial coefficients. The conjugate
P
Cesàro means T̃nα of order r of
an will be defined by
T̃nα =
S̃nα
.
Aα
n
The following formulae will also be needed:
α+1
S̃nα+1 − S̃n−1
S̃nα (S̃pr ) = S̃nα+r+1 ,
n
P
β
α+β+1
= S̃nα ,
Aα
.
n−p Ap = An
p=0
The differences of order α of the sequence {an } for any positive integer α are defined
by the equations 4α an = 4(4α−1 an ), n = 0, 1, 2, . . . 41 an = an − an+1 . Since
A−α−1
= 0 for m ≥ α + 1, we have
m
4α an =
α
P
m=0
−α−1
Am
an+m =
∞
P
m=0
A−α−1
an+m .
m
(2.1)
If the series (2.1) is convergent for some α which is not a positive integer, then we
denote the differences
4α an =
∞
P
m=0
−α−1
Am
an+m
n = 0, 1, 2, 3, . . . .
The broken differences 4α
n ap are defined by
4α
n ap =
n−p
P
m=0
A−α−1
ap+m .
m
By repeated partial summation of order r, we have
n
P
p=0
ap bp =
n
P
p=0
S̃pα−1 (ap )4α
n bp .
222
K. Singh, K. Kaur
3. Lemmas
We need the following lemmas for the proof of our result.
Lemma 3.1 [3] If r ≥ 0, p ≥ 0,
(i) ²n = o(n)−p , and
P∞
r+1
(ii) n=0 Ar+p
²n | < ∞,
n |4
then
(iii)
(iv)
n → ∞.
P∞
λ+p
|4λ+1 ²n | < ∞, for−1 ≤ λ ≤ r and
n=0 An
4λ ²n is of bounded variation for 0 ≤ λ
Aλ+p
n
≤ r and tends to zero as
Lemma 3.2 [1] Let r be a non-negative real number. If the sequence {²n }
satisfies the conditions
(i) ²n = O(1), and
∞
P
(ii)
nr |4r+1 ²n | < ∞,
n=1
β
then 4 ²n =
∞
P
m=0
r+1
²n+m , for β > 0.
Ar−β
m 4
Lemma 3.3 [2] If 0 < δ < 1 and 0 ≤ n < m, then
¯
¯
m
¯P
¯
¯ Aδ−1 Si ¯ ≤ max |Spδ |.
n−i ¯
¯
0≤p≤m
i=0
Lemma 3.4 [10] Let S̃n (x) and T̃nα be the nth partial sum and Cesàro mean of
order α > 0, respectively, of the series sin x + sin 2x + sin 3x + · · · + sin nx + · · · .
Then
Rπ
(i) 0 |S̃n (x)| dx ∼ log n,
Rπ
(ii) 0 |T̃nα | dx remains bounded for all n.
4. Main result
The main result of this paper is the following theorem.
Theorem 4.1. Let α be a positive real number. If a sequence {ak } belongs to
the class Kα , then for α ≤ r ≤ α + 1
(i) Knr (x) converges to K(x) pointwise for 0 < δ ≤ x ≤ π, and
(ii) Knr (x) → K(x) in the L1 -norm.
If we take α = 1 and r = 1, then Theorem 2 is obtained as a particular case
and also Theorem 4 can be deduced as a special case of Theorem 4.1 if we take
r = 1 in (1.5) .
On L1 -convergence of certain generalized modified trigonometric sums
223
Proof. We have
∞
P
1
(4r ak−1 − 4r ak+1 )S̃kr−1 (x)
2 sin x k=1
K(x) =
and
Knr (x) =
n
P
1
(4r ak−1 − 4r ak+1 )S̃kr−1 (x).
2 sin x k=1
Case 1. Let r = α + 1. Then
Knr (x) =
n
P
1
(4α+1 ak−1 − 4α+1 ak+1 )S̃kα (x).
2 sin x k=1
So Knr (x) converges to K(x) pointwise for 0 < δ ≤ x ≤ π.
Now
Z π
|K(x) − Knr (x)| dx
0
¯
Z π¯
∞
¯
¯ 1
P
α+1
α+1
α
¯
¯ dx
=
(4
a
−
4
a
)
S̃
(x)
k−1
k+1
k
¯
¯ 2 sin x
k=n+1
0
∞
Rπ
P
≤C
|(4α+1 ak−1 − 4α+1 ak+1 )| 0 |S̃kα (x)| dx
=C
k=n+1
∞
P
k=n+1
∞
P
α+1
Aα
ak−1 − 4α+1 ak+1 )|
k |(4
≤ C1
k=n+1
Rπ
0
|T̃kα (x)| dx
α+1
Aα
ak−1 − 4α+1 ak+1 )| = o(1), by hypothesis of Theorem 4.1.
k |(4
Therefore, Knr (x) converges to K(x) as n → ∞ in the L1 −norm.
Case 2. Let α < r < α + 1. Take r = α + 1 − δ and 0 < δ < 1. Then
n
P
1
(4r ak−1 − 4r ak+1 )S̃kr−1 (x)
2 sin x k=1
n
P
1
(4α+1−δ ak−1 − 4α+1−δ ak+1 )S̃kα−δ (x).
=
2 sin x k=1
Knr (x) =
Applying Abel’s transformation of order −δ and using Lemma 3.2, we have
n
P
1
(4α+1 ak−1 − 4α+1 ak+1 )S̃kα (x)
2 sin x k=1
· n
¸
n−k
P δ−1 α+1
P α−δ
1
α+1
=
S̃
(x)
Am (4
am+k−1 − 4
am+k+1 )
2 sin x k=1 k
m=1
· n
P α−δ
1
=
S̃
(x){(4α−δ+1 ak−1 − 4α−δ+1 ak+1 )
2 sin x k=1 k
¸
∞
P
δ−1
δ+1
δ+1
−
Am (4 am+k−1 − 4 am+k+1 )}
m=n−k+1
224
K. Singh, K. Kaur
·
¸
n
P
1
α−δ
α−δ+1
α−δ+1
=
S̃
(x)(4
ak−1 − 4
ak+1 ) − Rn (x) ,
2 sin x k=1 k
where
n
P
Rn (x) =
k=1
δ+1
S̃kα−δ (x){Aδ−1
an − 4δ+1 an+2 )
n−k+1 (4
δ+1
+ Aδ−1
an+1 − 4δ+1 an+3 ) + · · · }
n−k+2 (4
n
P
δ+1
=
S̃kr−δ (x){Aδ−1
an+2 − 4δ+1 an )
n−k+1 (4
k=1
n
P
+
k=1
This implies that
Knr (x) =
δ+1
S̃kr−δ (x)Aδ−1
an+3 − 4δ+1 an+1 ) + · · · }.
n−k+2 (4
· n
¸
P α
1
S̃k (x)(4α+1 ak−1 − 4α+1 ak+1 ) + Rn (x) .
2 sin x k=1
Hence
Z π
|K(x) − Knr (x)| dx
0
¯
Z π¯
∞
¯
¯ 1
P
α
α+1
α+1
¯ dx
¯
{
S̃
(x)(4
a
−
4
a
)
−
R
(x)}
≤C
k−1
k+1
n
k
¯
¯ 2 sin x
k=n+1
0
∞
Rπ
Rπ
P
≤C
|(4α+1 ak−1 − 4α+1 ak+1 )| 0 |S̃kα (x)| dx + 0 |Rn (x)| dx
=C
k=n+1
∞
P
α+1
Aα
ak−1 − 4α+1 ak+1 )|(x)
k |(4
k=n+1
∞
P
α+1
Aα
ak−1
k |(4
k=n+1
Z π
≤
= o(1) +
− 4α+1 ak+1 )| +
Rπ
0
Rπ
0
|T̃kα (x)| dx +
Rπ
0
|Rn (x)| dx
|Rn (x)| dx
|Rn (x)| dx, by hypotheses of Theorem 4.1.
0
Z
0
Now
π
|Rn (x)| dx
¶
Z π ¯µ n
¯ P α−δ
δ−1
δ+1
¯
=
S̃
(x)A
an − 4δ+1 an+2 )
k
n−k+1 (4
¯
k=1
0
¯
¶
µ n
¯
P α−δ
δ−1
δ+1
δ+1
S̃k (x)An−k+2 (4 an+1 − 4 an+3 ) + · · · ¯¯ dx
+
k=1
Z
π
≤
0
¯X
¯
¯ n α−δ
¯
¯
|(4δ+1 an − 4δ+1 an+2 )| ¯¯
S̃k (x)Aδ−1
n−k+1 ¯ dx
Z
+
k=1
π
|(4
0
δ+1
an+1 − 4
δ+1
¯ n
¯
¯ P α−δ
¯
δ−1
¯
an+3 )| ¯
S̃k (x)An−k+2 ¯¯ dx + · · ·
k=1
(4.1)
On L1 -convergence of certain generalized modified trigonometric sums
Z
225
π
|(4δ+1 an − 4δ+1 an+2 )| max |S̃pδ (x)| dx
0≤p≤n+1
Z π
+
|(4δ+1 an+1 − 4δ+1 an+3 )| max |S̃pδ (x)| dx + · · · , by Lemma 3.3
0≤p≤n+2
0
Z π
= |(4δ+1 an − 4δ+1 an+2 )|Aδn+1
max |T̃pδ (x)| dx
0≤p≤n+1
0
Z π
δ+1
δ+1
+ |(4 an+1 − 4 an+3 )|Aδn+2
max |T̃pδ (x)| + · · ·
≤
0
=
CAδn+1 |(4δ+1 an
δ+1
−4
an+2 )| +
0 0≤p≤n+2
δ
CAn+2 |(4δ+1 an+1
− 4δ+1 an+3 )| + · · ·
= o(1), by Lemmas 3.1 and 3.4.
Thus
Z
π
|Rn (x)| dx = o(1),
n → ∞.
0
Hence by (4.1)
Z
π
lim
n→∞
|K(x) − Knr (x)| dx = o(1).
0
Case 3. Let α = r. In this case
Knr (x) =
n
P
1
(4α ak−1 − 4α ak+1 )S̃kα−1 (x).
2 sin x k=1
Applying Abel’s transformation, we have
i
n
1 hP
Knr (x) =
(4α+1 ak−1 − 4α+1 ak+1 )S̃kα (x) + (4α an − 4α an+2 )S̃nα (x) .
2 sin x k=1
Since S̃kα (x) are bounded for 0 < δ ≤ x ≤ π,
Knr (x) → K(x) =
∞
P
1
(4α+1 ak−1 − 4α+1 ak+1 )S̃kα (x)
2 sin x k=1
pointwise for 0 < δ ≤ x ≤ π. So
Z π
|K(x) − Knr (x)| dx
0
¯
Z π¯
∞
¯ 1
¯
P
α+1
α+1
α
α
α
α
¯
=
(4
ak−1 − 4
ak+1 )S̃k (x) − (4 an − 4 an+2 )S̃n (x)¯¯ dx
¯ 2 sin x
k=n+1
0
Z π
∞
P
|(4α+1 ak−1 − 4α+1 ak+1 )| |S̃kα (x)| dx
≤C
k=n+1
Z π
+
0
=
∞
P
k=n+1
0
|(4α an − 4α an+2 )| |S̃nα (x)| dx
Aα
k
Rπ
0
|(4α+1 ak−1 − 4α+1 ak+1 )||T̃kα (x)| dx
226
K. Singh, K. Kaur
Z
α
α
+ Aα
n |(4 an − 4 an+2 )|
≤C
∞
P
k=n+1
π
0
|T̃nα (x)| dx
α+1
Aα
ak−1 − 4α+1 ak+1 )| + C1 |(4α an − 4α an+2 )|
k |(4
= o(1) + o(1) = o(1), by the hypotheses of Theorem 4.1 and Lemma 3.1.
Thus the proof is complete.
REFERENCES
[1] Andersen, A.F., On extensions within the theory of Cesàro summability of a classical 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), 166–173.
[3] Bosanquet, L.S., Note on convergence and summability factors (III), Proc. London Math.
Soc. (1949), 482–496.
[4] Garrett, J.W., Stanojević, Č.V., On L1 -convergence of certain cosine sums, Proc. Amer.
Math. Soc. 54 (1976), 101–105.
[5] Kaur, K., Bhatia S.S., Integrability and L1 -convergence of Rees-Stanojević sums with generalized semi-convex coefficients, Int. J. Math. and Math. Sci. 30(11) (2002), 645–650.
[6] Kaur, K., Bhatia, S.S., and Ram, B., On L1 -convergence of certain trigonometric sums,
Georgian Math. J. 1(11) (2004), 98–104.
[7] Kaur, K., On L1 -convergence of certain trigonometric sums with generalized sequence K α ,
Kyungpook Math. J. 45 (2004), 73–85.
[8] Kolmogorov, A.N., Sur l’ordre de grandeur des coefficients de la series de Fourier-Lebesque,
Bull. Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1923), 83–86.
[9] Teljakovskii, S.A., Some bounds for trigonometric series with quasi-convex coefficients, Mat.
Sb. 63(105) (1964), 426–444.
[10] Zygmund, A., Trigonometric Series, Vol. 1, 2, Cambridge University Press, 1959.
(received 16.06.2008)
Department of Applied Sciences, GZS College of Engineering and Technology, Bathinda-151001,
Punjab, India
E-mail: karanvir786@indiatimes.com, mathkaur@yahoo.co.in