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J. Nonlinear Sci. Appl. 7 (2014), 272–277
Research Article
Certain sufficient conditions on |N, pn, qn|k
summability of orthogonal series
Xhevat Z. Krasniqi
Department of Mathematics and Informatics, Faculty of Education, University of Prishtina ”Hasan Prishtina”, Avenue ”Mother
Theresa” 5, 10000 Prishtinë, Kosovë.
Communicated by Mohd Salmi Md Noorani
Abstract
In this paper we obtain some sufficient conditions on |N, pn , qn |k summability of an orthogonal series. These
conditions are expressed in terms of the coefficients of the orthogonal series. Also, several known and new
c
results are deduced as corollaries of the main results. 2014
All rights reserved.
Keywords: Orthogonal series, generalized Nörlund summability.
2010 MSC: 42C15, 40F05, 40G05.
1. Introduction
P
Let ∞
n=0 an be a given infinite series with its partial sums {sn }. Then, let p denotes the sequence {pn }.
For two given sequences p and q, the convolution (p ∗ q)n is defined by
(p ∗ q)n =
n
X
pm qn−m =
m=0
We write
Rn := (p ∗ q)n , Rnj :=
n
X
n
X
pn−m qm
m=j
and
Rnn+1 = 0, Rn0 = Rn .
Email address: xhevat.krasniqi@uni-pr.edu (Xhevat Z. Krasniqi)
Received 2010-9-29
pn−m qm .
m=0
Xh. Z. Krasniqi, J. Nonlinear Sci. Appl. 7 (2014), 272–277
Also we put
Pn := (p ∗ 1)n =
n
X
and Qn := (1 ∗ q)n =
pm
m=0
273
n
X
qm .
m=0
When Rn 6= 0 for all n, the generalized Nörlund transform of the sequence {sn } is the sequence {tp,q
n }
obtained by putting
n
1 X
tp,q
=
pn−m qm sm .
n
Rn
m=0
P
The infinite series ∞
n=0 an is said to be absolutely summable (N, pn , qn )k with index k, if for k ≥ 1 the
series
∞ X
Rn k−1 p,q
k
|tn − tp,q
n−1 |
qn
n=0
converges [8], and we write in brief
∞
X
an ∈ |N, pn , qn |k .
n=0
We note that for k = 1, |N, pn , qn |k summability is the same as |N, pn , qn | ≡ |N, pn , qn |1 summability
introduced by Tanaka [7].
Let {ϕn (x)} be an orthonormal system defined in the interval (a, b). We assume that f (x) belongs to
L2 (a, b) and
∞
X
f (x) ∼
an ϕn (x),
(1.1)
n=0
Rb
where an = a f (x)ϕn (x)dx, (n = 0, 1, 2, . . . ).
Our main purpose of the present paper is to study the |N, pn , qn |k summability of the orthogonal series
(1.1), for 1 ≤ k ≤ 2, and to deduce as corollaries all results of Y. Okuyama [6].
Throughout this paper K denotes a positive constant that it may depends only on k, and be different
in different relations.
The following lemma due to Beppo Levi (see, for example [3]) is often used in the theory of functions in
which are involved the series and integrals and which are involved in [1] and [2] too. It will need us to prove
main results.
Lemma 1.1. If fn (t) ∈ L(E) are non-negative functions and
∞ Z
X
fn (t)dt < ∞,
(1.2)
n=1 E
then the series
∞
X
fn (t)
n=1
converges almost everywhere on E to a function f (t) ∈ L(E). Moreover, the series (1.2) is also convergent
to f in the norm of L(E).
2. Main results
We prove the following theorem.
Xh. Z. Krasniqi, J. Nonlinear Sci. Appl. 7 (2014), 272–277
274
Theorem 2.1. If for 1 ≤ k ≤ 2 the series
∞
X
(
n=0
Rn
qn
2− 2 X
n
k
j
Rn−1
Rnj
−
Rn Rn−1
j=1
converges, then the orthogonal series
∞
X
)k
!2
2
|aj |2
an ϕn (x)
n=0
is summable |N, pn , qn |k almost everywhere.
Proof. For the generalized Nörlund transform tp,q
n (x) of the partial sums of the orthogonal series
we have that
tp,q
n (x)
=
P∞
n=0 an ϕn (x)
n
m
X
1 X
pn−m qm
aj ϕj (x)
Rn
m=0
n
X
j=0
n
X
j=0
m=j
=
1
Rn
=
n
1 X j
Rn aj ϕj (x)
Rn
aj ϕj (x)
pn−m qm
j=0
Pm
where j=0 aj ϕj (x) are partial sums of order k of the series (1.1).
As in [6] page 163 one can find that
p,q
p,q
4tp,q
n (x) := tn (x) − tn−1 (x) =
n
X
j=1
Rj
Rnj
− n−1
Rn Rn−1
!
aj ϕj (x).
Using the Hölder’s inequality and orthogonality to the latter equality, we have that
k2
Z b
Z b
p,q
1− k2
2
p,q
p,q
k
|4tn (x)| dx ≤ (b − a)
|tn (x) − tn−1 (x)| dx
a
a
1− k2
= (b − a)
"
n
X
j=1
Rj
Rnj
− n−1
Rn Rn−1
#k
!2
2
|aj |2
.
Hence, the series
Z
∞ X
Rn k−1
n=1
qn
b
k
|4tp,q
n (x)| dx ≤ K
a
" n
∞ X
Rn k−1 X
n=1
qn
j=1
Rj
Rnj
− n−1
Rn Rn−1
#k
!2
2
|aj |2
(2.1)
converges by the assumption. From this fact and since the functions |4tp,q
n (x)| are non-negative, then by
the Lemma 1.1 the series
∞ X
Rn k−1
k
|4tp,q
n (x)|
qn
n=1
converges almost everywhere. This completes the proof of the theorem.
For k = 1 in Theorem 2.1 we have the following result.
Xh. Z. Krasniqi, J. Nonlinear Sci. Appl. 7 (2014), 272–277
275
Corollary 2.2 ([6]). If the series
∞
X
(
n=0
n
X
j=1
converges, then the orthogonal series
j
Rn−1
Rnj
−
Rn Rn−1
∞
X
)1
!2
2
|aj |2
an ϕn (x)
n=0
is summable |N, pn , qn | almost everywhere.
Let us prove now another two corollaries of the Theorem 2.1.
Corollary 2.3. If for 1 ≤ k ≤ 2 the series
!k (
∞
X
pn
n=0
Pn Pn−1
n
X
1/k
p2n−j
j=1
converges, then the orthogonal series
∞
X
Pn Pn−j
−
pn
pn−j
2
)k
2
|aj |2
an ϕn (x)
n=0
is summable |N, pn |k ≡ |N, pn , 1|k almost everywhere.
Proof. After some elementary calculations one can show that
Rj
Rnj
pn
− n−1 =
Rn Rn−1
Pn Pn−1
Pn Pn−j
−
pn
pn−j
pn−j
for all qn = 1, and the proof follows immediately from Theorem 2.1.
Remark 2.4. We note that:
1. If pn = 1 for all values of n then |N, pn |k summability reduces to |C, 1|k summability
2. If k = 1 and pn = 1/(n + 1) then |N, pn |k is equivalent to |R, log n, 1| summability.
These facts show us that Theorem 2.1 includes also sufficient conditions under which the series (1.1) is
|C, 1|k summable, respectively |R, log n, 1| summable.
Corollary 2.5. If for 1 ≤ k ≤ 2 the series
∞
X
!k (
1/k
qn
n
X
1/k
n=0
converges, then the orthogonal series
Qn Qn−1
∞
X
)k
2
Q2j−1 |aj |2
j=1
an ϕn (x)
n=0
is summable |N , qn |k ≡ |N , 1, qn |k almost everywhere.
Proof. Since
j
Rn−1
qn Qj−1
Rnj
−
=−
Rn Rn−1
Qn Qn−1
for all pn = 1, then the proof follows directly from Theorem 2.1.
Xh. Z. Krasniqi, J. Nonlinear Sci. Appl. 7 (2014), 272–277
276
Also, putting k = 1 in Corollaries 2.3 and 2.5 we obtain
Corollary 2.6 ([4]). If the series
∞
X
n=0
pn
Pn Pn−1
(
n
X
p2n−j
j=1
converges, then the orthogonal series
∞
X
Pn Pn−j
−
pn
pn−j
2
)1
2
|aj |2
an ϕn (x)
n=0
is summable |N, pn | almost everywhere.
Corollary 2.7 ([5]). If the series
∞
X
n=0
qn
Qn Qn−1
converges, then the orthogonal series
(
∞
X
n
X
)1
2
Q2j−1 |aj |2
j=1
an ϕn (x)
n=0
is summable |N , qn | almost everywhere.
Now we shall prove a general theorem concerning |N, pn , qn |k summability of an orthogonal series which
involves a positive sequence with certain additional conditions.
For this reason first we put
!2
2− 2
∞
j
j
X
k
R
4
R
R
1
n
n
Λ(k) (j) := 2
− n−1
n k −2
(2.2)
qn
Rn Rn−1
j k −1
n=j
then the following theorem holds true.
Theorem 2.8. Let 1 ≤P
k ≤ 2 and {Ω(n)} be a positive sequence such that {Ω(n)/n} is a non-increasing
1
sequence and the series ∞
n=1 nΩ(n) converges. Let {pn } and {qn } be non-negative. If the series
∞
X
2
|an |2 Ω k −1 (n)Λ(k) (n)
n=1
converges, then the orthogonal series
defined by (2.2).
P∞
n=0 an ϕn (x)
∈ |N, pn , qn |k almost everywhere, where Λ(k) (n) is
Proof. Applying Hölder’s inequality to the inequality (2.1) we get that
Z
∞ X
Rn k−1
n=1
≤K
qn
" n
∞ X
Rn k−1 X
qn
∞
X
n=1
≤K
k
|4tp,q
n (x)| dx ≤
a
n=1
=K
b
j=1
"
1
(nΩ(n))
∞
X
n=1
Rj
Rnj
− n−1
Rn Rn−1
2−k
2
1
(nΩ(n))
(nΩ(n))
! 2−k "
2
∞
X
n=1
2
−1
k
Rn
qn
(nΩ(n))
#k
!2
2
|aj |2
2− 2 X
n
k
2
−1
k
j=1
Rn
qn
Rj
Rnj
− n−1
Rn Rn−1
2− 2 X
n
k
j=1
#k
!2
2
|aj |2
j
Rn−1
Rnj
−
Rn Rn−1
#k
!2
2
|aj |2
Xh. Z. Krasniqi, J. Nonlinear Sci. Appl. 7 (2014), 272–277
(
≤K
(
≤K
∞
X
2
|aj |
∞
X
j=1
n=j
∞
X
|aj |2
j=1
(
=K
∞
X
(nΩ(n))
Ω(j)
j
2
−1
k
2 −1 X
∞
k
n
Rn
qn
4
−2
k
n=j
2− 2
k
Rn
qn
277
j
Rn−1
Rnj
−
Rn Rn−1
2− 2
k
!2 ) k
Rj
Rnj
− n−1
Rn Rn−1
2
!2 ) k
2
)k
|aj |2 Ω
2
−1
k
2
(j)Λ(k) (j)
,
j=1
which is finite by assumption, and this completes the proof.
A direct consequence of the theorem 2.8 is the following (k = 1).
CorollaryP
2.9 ([6]). Let {Ω(n)} be a positive sequence such that {Ω(n)/n} is a non-increasing
sequence and
P∞
1
2 Ω(n)Λ(1) (n)
|a
|
the series ∞
converges.
Let
{p
}
and
{q
}
be
non-negative.
If
the
series
n
n
n=1 nΩ(n)
n=1 n
P
a
ϕ
(x)
∈
|N,
p
,
q
|
almost
everywhere,
where
Λ(1) (n) is defined
converges, then the orthogonal series ∞
n n
n=0 n n
2
j
j
P
2 Rn − Rn−1
by Λ(1) (j) := 1j ∞
n
.
n=j
Rn
Rn−1
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