KRASNIQI, XHEVAT

English

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ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 2 (2013), no.1, 916
ISSN 1857-8365, UDC: 517.57:517.521
ON A PRODUCT SUMMABILITY OF AN ORTHOGONAL SERIES
XHEVAT Z. KRASNIQI
Abstract. In this paper we have dened a new product summability, in order to
make an advanced study in the special topic of summability. Namely, we give some
sucient conditions, in terms of the coecients of an orthogonal series, under which
such series is product summable almost everywhere.
1. Introduction
The absolute summability is a generalization of the concept of the absolute convergence
just as the summability is an extension of the concept of the convergence. There are a
lot of notions of absolute summability dened by several authors. Particularly, by those
authors such notions are employed for studying the absolute summability of an orthogonal
series. As a recent result can be mentioned those of Y. Okuyama (see section 2) who
has proved two theorems which give sucient conditions in terms of the coecients
of an orthogonal series under which such series would be absolute generalized Nörlund
summable almost everywhere. Moreover, an interested reader could nd some new results,
see as examples [4]-[6], where are given some statements which include all of the results
previously proved by Y. Okuyama and T. Tsuchikura [8]-[9], and also are given some new
consequences. In order to make an advance study in this direction, here we study the
question when an orthogonal series is product summable almost everywhere.
2. Notations and Known Results
For two sequences of real or complex numbers fpng and fqng, let
Pn = p0 + p1 + p2 +   + pn =
nX
v=0
pv;
Qn = q0 + q1 + q2 +   + qn =
nX
v=0
qv;
2010 Mathematics Subject Classication. 42C15, 40F05, 40G05.
Key words and phrases. Orthogonal series, absolute summability, Fourier series.
9
10 X. Z. KRASNIQI
and let the convolution (p  q)n be dened by
Rn := (p  q)n :=
nX
v=0
pvqn v; and denote R
j
n :=
nX
v=j
pvqn v:
Let
P
1
n=0 an be a given innite series with the sequence of its n th partial sums fsng.
We write
tp;qn =
1
Rn
nX
v=0
pn vqvsv:
If Rn 6= 0 for all n, the generalized Nörlund transform of the sequence fsng is the sequence
ft
p;q
n g.
The innite series
P
1
n=0 an is said to be absolutely summable (N; p; q) if the series
1X
n=1
jtp;qn   t
p;q
n 1j
converges, and we write in brief
1X
n=0
an 2 jN; p; qj:
The jN; p; qj summability was introduced by Tanaka [3].
Let f'j(x)g be an orthonormal system dened in the interval (a; b). We assume that
f belongs to L2(a; b) and
(2.1) f(x) 
1X
j=0
cj'j(x);
where cj =
R b
a
f(x)'j(x)dx; (j = 0; 1; 2; : : : ).
Regarding to the orthogonal series (2.1) Y. Okuyama has proved the following two
theorems:
Theorem 2.1 ([8]). If the series
1X
n=1
(
nX
j=1
 
R
j
n
Rn
 
R
j
n 1
Rn 1
!2
jcj j
2
) 1
2
converges, then the orthogonal series
1X
j=0
cj'j(x)
is summable jN; p; qj almost everywhere.
Theorem 2.2 ([8]). Let f
(n)g be a positive sequence such that f
(n)=ng is a non-
increasing sequence and the series
1X
n=1
1
n
(n)
ON A PRODUCT SUMMABILITY OF AN ORTHOGONAL SERIES 11
converges. Let fpng and fqng be non-negative. If the series
1X
n=1
jcnj
2
(n)w(n)
converges, then the orthogonal series
1X
j=0
cj'j(x) 2 jN; p; qj
almost everywhere, where w(n) is dened by
w(j) := j 1
1X
n=j
n2
 
R
j
n
Rn
 
R
j
n 1
Rn 1
!2
:
If we take pv = 1 for all v then, the sequence-to-sequence transformation t
p;q
n reduces
to transformation Rqn :=
1
Qn
nX
v=0
qvsv, while for qv = 1 we obtain the transformation
Rpn :=
1
Pn
nX
v=0
pn vsv. G. Das [2] dened the transformation
Un :=
1
Pn
nX
v=0
pn v
Qv
vX
j=0
qv jsj ;
and gave the following denition:
Denition 2.1. The innite series
P
1
n=0 an is said to be summable j(N; p)(N; q)j, if
the sequence fUng is of bonded variation, i.e. the series
1X
n=1
jUn   Un 1j
converges.
Later on, W. T. Sulaiman [1] considered the transformation
Vn :=
1
Qn
nX
v=0
qv
Pv
vX
j=0
pjsj
of the sequence fsng, and presented the denition:
Denition 2.2. The innite series
P
1
n=0 an is said to be summable j(R; qn)(R; pn)jk,
k  1, if
1X
n=1
nk 1jVn   Vn 1j
k
converges, and we write in brief
1X
n=0
an 2 j(R; qn)(R; pn)jk:
12 X. Z. KRASNIQI
Let us denote by Dn the transformation
Dn :=
1
Rn
nX
v=0
pn vqv
Rv
vX
j=0
pjqv jsj
of the sequence fsng.
Now we shall introduce the following denition:
Denition 2.3. The innite series
P
1
n=0 an is said to be summable j(N; pn; qn)
(N; qn; pn)jk, k  1, if
1X
n=1
nk 1jDn  Dn 1j
k
converges, and we write shortly
P
1
n=0 an 2 j(N; pn; qn)(N; qn; pn)jk:
The main purpose of the present paper is to study the j(N; pn; qn)(N; qn; pn)jk summa-
bility of the orthogonal series (2.1) for 1  k  2.
Throughout K denotes a positive constant that it may depends only on k, and be
dierent in dierent relations.
The following lemma due to Beppo Levi (see, for example [7]) is often used in the
theory of functions. It will need us to prove main results.
Lemma 2.1. If fn(t) 2 L(E) are non-negative functions and
(2.2)
1X
n=1
Z
E
fn(t)dt <1;
then the series
1X
n=1
fn(t)
converges almost everywhere on E to a function f(t) 2 L(E). Moreover, the series
(2.2) is also convergent to f in the norm of L(E).
3. Main Results
We prove the following theorem.
Theorem 3.1. If for 1  k  2 the series
1X
n=1
24n2  2k nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
jcij
2
35
k
2
converges, then the orthogonal series
1X
n=0
cn'n(x)
is summable j(N; pn; qn)(N; qn; pn)jk almost everywhere.
ON A PRODUCT SUMMABILITY OF AN ORTHOGONAL SERIES 13
Proof. First we consider the case k 2 (1; 2). We use the notations
eRin := nX
v=i
qn vpv
Rv
; eRnn 1 = 0:
Let
sj(x) =
jX
i=0
ci'i(x)
be the partial sums of order j of the series (2.1). Then, for the transform Dn(x) of the
partial sums sj(x), we have
Dn(x) =
1
Rn
nX
v=0
pn vqv
Rv
vX
j=0
pjqv j
jX
i=0
ci'i(x)
=
1
Rn
nX
v=0
pn vqv
Rv
vX
i=0
ci'i(x)
vX
j=i
pjqv j
=
1
Rn
nX
v=0
pn vqv
Rv
vX
i=0
Rivci'i(x)
=
1
Rn
nX
i=0
Rinci'i(x)
nX
v=i
pn vqv
Rv
=
nX
i=0
Rin eRin
Rn
ci'i(x):
Whence,
4Dn(x) = Dn(x) Dn 1(x)
=
nX
i=0
Rin eRin
Rn
ci'i(x) 
n 1X
i=0
Rin 1 eRin 1
Rn 1
ci'i(x)
=
nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!
ci'i(x):
Using the Hölder's inequality and orthogonality to the latter equality, we obtainZ b
a
j4Dn(x)j
kdx  (b  a)1 
k
2
 Z b
a
jDn(x) Dn 1(x)j
2dx
! k
2
= (b  a)1 
k
2
24 nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
jcij
2
35
k
2
:
Subsequently, the series
(3.1)
1X
n=1
nk 1
Z b
a
j4Dn(x)j
kdx  K
1X
n=1
nk 1
24 nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
jcij
2
35
k
2
converges, since the last one does. Now according to the Lemma 2.1 the series (2.1)
is summable j(N; pn; qn)(N; qn; pn)jk almost everywhere. For k = 2 we apply only the
14 X. Z. KRASNIQI
orthogonality, as far as for k = 1 we apply the well-known Schwarz's inequality. This
completes the proof of the theorem. 
Now we shall prove the counterpart of Theorem 3.1 (it can be seen also as the coun-
terpart of a theorem of P. L. Ul'yanov [10]). It is a general theorem which involves in it
a new positive sequence with some additional conditions. For this reason rst we put:
(3.2) R(k)(i) :=
1
i
2
k
 1
1X
n=i
n
2
k
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
;
and then the following theorem holds true.
Theorem 3.2. Let 1  k  2 and f
(n)g be a positive sequence such that f
(n)=ng
is a non-increasing sequence and the series
P
1
n=1
1
n
(n) converges.
Let fpng and fqng be non-negative. If the series
1X
n=1
jcnj
2

2
k
 1(n)R(k)(n)
converges, then the orthogonal series
1X
n=0
cn'n(x) 2 j(N; pn; qn)(N; qn; pn)jk
almost everywhere, where R(k)(n) is dened by (3.2).
Proof. Applying Hölder's inequality to the inequality (3.1) we get that
1X
n=1
nk 1
Z b
a
j4Dn(x)j
kdx
 K
1X
n=1
nk 1
24 nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
jcij
2
35
k
2
= K
1X
n=1
1
(n
(n))
2 k
2
24(n
(n)) 2k 1 n2  2k nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
jcij
2
35
k
2
 K
 
1X
n=1
1
n
(n)
! 2 k
2
24 1X
n=1
(n
(n))
2
k
 1
n2 
2
k
nX
i=1
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!2
jcij
2
35
k
2
 K
8<:
1X
i=1
jcij
2
1X
n=i
(n
(n))
2
k
 1
n2 
2
k
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!29=;
k
2
 K
8<:
1X
i=1
jcij
2


(i)
i
 2
k
 1 1X
n=i
n
2
k
 
Rin eRin
Rn
 
Rin 1 eRin 1
Rn 1
!29=;
k
2
 K
(
1X
i=1
jcij
2

2
k
 1(i)R(k)(i)
) k
2
;
ON A PRODUCT SUMMABILITY OF AN ORTHOGONAL SERIES 15
which is nite by assumption. Doing the same reasoning as in the proof of Theorem 3.1
we easy arrive to nish the proof. 
It is obvious that the transformation Dn can never be the same as 
p;q
n , therefore
Theorems 3.1-3.2 bring new results. Moreover, transformation Dn can not reduces to
the transformations Un or Vn, i.e. it would be of particular interest to answer ques-
tions: Under what conditions an orthogonal series of the form (2.1) is (N; p)(N; q) or
(R; qn)(R; pn)k summable? Regarding to these questions, in the following, we shall give
four theorems without their proofs.
Denote
eP i;qn = nX
v=i
pn v
Qv
; eQi;pn = nX
v=i
qv
Pv
; and ePn;qn 1 = eQn;pn 1 = 0:
Theorem 3.3. If the series
1X
n=1
24 nX
i=1
 
Qin eP i;qn
Pn
 
Qin 1 eP i;qn 1
Pn 1
!2
jcij
2
35
1
2
converges, then the orthogonal series
1X
n=0
cn'n(x)
is summable j(N; p)(N; q)j almost everywhere.
Theorem 3.4. Let f
(n)g be a positive sequence such that f
(n)=ng is a non-
increasing sequence and the series
P
1
n=1
1
n
(n) converges.
Let fpng and fqng be non-negative. If the series
1X
n=1
jcnj
2
(n)Rp;q(n)
converges, then the orthogonal series
1X
n=0
cn'n(x) 2 j(N; p)(N; q)j
almost everywhere, where Rp;q(n) is dened by
Rp;q(i) :=
1
i
1X
n=i
n2
 
Qin eP i;qn
Pn
 
Qin 1 eP i;qn 1
Pn 1
!2
:
Theorem 3.5. If for 1  k  2 the series
1X
n=1
24n2  2k nX
i=1
 
P in eQi;pn
Qn
 
P in 1 eQi;pn 1
Qn 1
!2
jcij
2
35
k
2
converges, then the orthogonal series
1X
n=0
cn'n(x)
16 X. Z. KRASNIQI
is summable j(R; qn)(R; pn)jk almost everywhere.
Theorem 3.6. Let 1  k  2 and f
(n)g be a positive sequence such that f
(n)=ng
is a non-increasing sequence and the series
P
1
n=1
1
n
(n) converges.
Let fpng and fqng be non-negative. If the series
1X
n=1
jcnj
2

2
k
 1(n)bR(p;q;k)(n)
converges, then the orthogonal series
1X
n=0
cn'n(x) 2 j(R; qn)(R; pn)jk
almost everywhere, where bR(p;q;k)(n) is dened by
bR(p;q;k)(i) := 1
i
2
k
 1
1X
n=i
n
2
k
 
P in eQi;pn
Qn
 
P in 1 eQi;pn 1
Qn 1
!2
:
Acknowledgment
The author would like to thank the anonymous referee for her/his advices which im-
proved the the nal form of this paper.
References
[1] W. T. Sulaiman: Note on product summability of an innite series, Int. J. Math. Math. Sci.
(2008), Art. ID 372604, 10 pp.
[2] G. Das: Tauberian theorems for absolute Nörlund summability, Proc. London Math. Soc. vol. 19,
no. 2 (1969), 357384.
[3] M. Tanaka: On generalized Nörlund methods of summability, Bull. Austral. Math. Soc., vol. 19,
(1978), 381402.
[4] Xh. Z. Krasniqi: An application of q-mathematics on absolute summability of orthogonal series,
Analysis (Munich) 33, 3543 (2013)/ DOI 10.1524/anly.2013.1169.
[5] Xh. Z. Krasniqi: On jA; jk-summability of orthogonal series, Math. Bohem., 137, (2012), 1725.
[6] Xh. Z. Krasniqi: On absolute weighted mean summability of orthogonal series, Selçuk J. Appl.
Math., 12 (2011), 6370.
[7] I. P. Natanson: Theory of functions of a real variable, Frederick Ungar New York 1955, 1961,
MR 16804, 26 # 6309.
[8] Y. Okuyama: On the absolute generalized Nörlund summability of orthogonal series, Tamkang J.
Math., 3 (2002), 161165.
[9] Y. Okuyama and T. Tsuchikura: On the absolute Riesz summability of orthogonal series,
Anal. Math., 7 (1981), 199208.
[10] P. L. Ul'yanov: Solved and unsolved problem in the theory of trigonometric and orthogonal series,
Uspekhi Mat. Nauk, 19 (1964), 369.
University of Prishtina
Faculty of Education
Department of Mathematics and Informatics
Avenue "Mother Theresa" 5
10000 Prishtina
Kosova
E-mail address: xhevat.krasniqi@uni-pr.edu


ON A PRODUCT SUMMABILITY OF AN ORTHOGONAL SERIES

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ON A PRODUCT SUMMABILITY OF AN ORTHOGONAL SERIES

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