Internat. J. Math. & Math. Sci.
VOL. ii NO. 3 (1988) 517-522
517
ON TRANSLATIVITY OF ABSOLUTE ROGOSlNSKI-
BERNSTEIN SUMMABILITY METHODS
AZMI K. AL-BIADI
Department of Mathematics
U.A.E. University, Ai-Ain
United Arab Emirates
(Received November 26, 1986 and in revised form February 25, 1987)
ABSTRACT.
The paper is concerned with the problem of translativity of an absolute
Rogosinski-Bernsteln summabillty methods
< h
results that when h;
Bhl
I,
IBhl
=<
0
h _<- I.
The paper contains two
is not translative neither to the left nor to the right.
KEYS WORDS AND PHRASES.
Absolute summabillty, Rogoslnskl-Bernsteln methods, sequence-
to sequence transformation,
absolutely regular, translatlve.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
(Bh);
is translatlve and when h; (o < h < ), then
Primary 40F05, Secondary 40G99.
INTRODUCTION.
A series
+
Sn a0 al
of real or complex terms a
a0+al+..,
+
order h; (0
+
h
an,
I), if t n
n
t
n
[
with its partial sums
k
is evaluable to t by the Rogoslnski-Bernstein method (B
[cos
k=0
/
,
t as n
wk
of
where
ak
2(n+h)’
The series is evaluable to M by the
h)
Ceasa’ro mean
of (C,I), if M
n
M as n
,
where
n
M
n
n+l
[
k--0
(1.2)
Sk"
The series is absolutely summable (B
h)
or summable
IBhl
if the sequence
{tn
is of
bounded variation; that is to say, if
0(I),
t_l
o.
(1.3)
Similar definition for the series being absolutely summable (C,I).
A sequence-to-sequence method A is called translative to the left, if the limitability of
S0,S
S
implies the llmitabillty of 0,S
n
0
limit, it is translative to the right, if the converse holds.
and only if, A is translative to the left and right.
Sn_ I,
to the same
A is translatlve if
A.K. AL-MADI
518
The method (B
h)
has been the subject of many papers (see [2], [3] and [4]).
Agnew [3] obtained many significant results concerning inclusion and equivalence
relations between the methods
(Bh),
[I; Theorems (3. I) and (3.2)
obtained results involving absolute summability methods
(C,I) and modified arithmetic means.
The author
analogous to some of those by Agnew.
h; 0
(Bh);
2.
I, for which (Bh)
h
3(/3-1) =<
h <
In [2], the author investigated the values of
is or is not translatlve, and left an open problem that
is non-translative to the left nor do to the right.
OBJECT OF THE PAPER.
The object of this paper is to obtain results on translatlvlty involving
absolute summabillty of the method
(Bh).
Some of these results are analogous to some
of those by the author in [2].
3.
SUBSIDIARY RESULTS.
The following results are necessary for our purposes:
THEOREM I.
(The author [i; Lemma 2.2]) A necessary condition that the sequence-
to-sequence transformation
gn
Gn,kSk
k=O
be absolutely regular is that G
n,k
be bounded for all n k
(The author [I; Theorem 3.2])
THEOREM 2.
If
< h &
I, then
IBhl
and
IC,ll
are
equivalent.
4.
MAIN RESULTS.
In this section we prove the two main results of this paper.
translative.
IBhl
IC,II is translative, Theorem 2 yields the result.
the left nor to the
(0,1/2), then IBhl is not translative
If h; h
THEOREM 3.
PROOF.
I, then
< h
If
is
Since
to
THEOREM 4.
right.
In (I.I) write for a k its value
PROOF.
sequence {Un
n
n
with
S_I
0, and consider the
given by
Un S0 + S1
to obtain t
Sk-Sk_ I,
+
in terms of U
+
n
sn
The result is
k= 0
where
A
n,n
(4.2)
sinh8
A
sin(l+h)O
nn-I
A
-4sin2 8cos (k+l) 8,
A
0
n,k
(4.3)
2slnO,
O=<k=<n-2
(4.4)
and
n,k
otherwise,
(4.5)
519
TRANSLATIVlTY OF ABSOLUTE ROGOSINSKI-BERNSTEIN SUMMABILITY METHODS
and where
(4.6)
2 (n+h)
Let
{tn }’ {n
(Bh)
be respectively the
in terms of t
transforms of
{Sn }’ {Sn-I }"
Obtain
-tn+
The result is
n
(4.7)
n’ktk’
kO
where
n
vn,k
0.<k.<n
v+IBv,k
v=k
(4.8)
and
V
n,k
and where
0
otherwise
(4.9)
is given by (4.2)-(4.5), and B
is the reciprocal matrix of
n,k
Using the inversion formula of (4.10, we have for each r=3,4,5
B
An,k
(-1)
n,n-r
H
n,n-r
"’’An-r,n-r
r
A
n,
An,k.
(4.10)
nAn_l,n
where
An_r+l,n_ r
0
0
0
An-r+2,n-r+2
0
0
An_r+l,n_r+
An-r+2,n-r An-r+2,n-r+l
(4.ii)
A
n,n-r
.
An,n-I
A
n,n-r+l
Using (4.8),(4.9) and the fact that
n-I
Bn,k =-(
A
(An, n)
B
n,v v,k
v=k
-I
(4.12)
we have
n-I
E
V
nk
n,v
B
0
vk
=<
(4.13)
k=< n-I
where
E
An+l ,n+l
n,v
An,n
A
n,v
+
(4.14)
An+l,v+l
Hence
r
V
n,n-r
E
u=l
B
n,n-u n-u,n-r
Let w be any constant with 0 < w < 1/2.
w
sufficiently large n, and 2 -<- r -<- n
r
<=n
We will show that if h; 0 < h <
(4.15)
,
then for
A.K. AL-MADI
520
(4.16)
where C is some positive constant less that I, and this will be satisfies if, we
r & nw) the terms of the sum of the
prove that (for sufficiently large n and 2
right hand side of (4.15) alternate in sign and
En,n-u-I Bn-u-I ,n-r
E
B
n,n-u
(4.17)
constant < I.
n-u,n-r
Using (4.2)-(4.4), we have that
nA
n,n
h
2
nAn
(l-h)
,n-I
An ,k <
2
0, with
A
n,k
0, (0-<k-<-n-2). (4.18)
Using (4.2)-(4.6) and (4.18), we have from (4.10) and (4.11) that
B
n,n-r
n
2
(-l)r
l-h r
()h
Thus
B
n-u,n-r
2
(-l)r-u
n
Next, we will show that E
l-h r-u
-<_ u
.
n
w.
r
n
(n-u+2)
4(n+h+l) cos 2(n+l+h)
sin2 4(n+h) cos (n-u+l)
2(n+h)
sin
A
n,n-u
as
h
> 0 for
n,n-u
(4.2) and (4.4) that if 2 -<_ u --< n, then
E
(4.19)
as
n,n-u
2
(4.20)
Using (4.14), we have from
sin
sin
2(n+l+h)
It may be easily seen that the quantity inside the square brackets is negative.
w
together with (4.4) imply that when 2 --< u -<- n
E
n,n-u
> 0
for large n uniformly in
2
w
u _-< n
(4.21)
2(n+h)
This
(4.22)
Using (4.2),(4.3) and note that
sin
wh
w3h3
wh
2(n+h)
2(n+h)
48(n+h)3
+ 0 __l5
(n
(4.23)
we have from (4.14) that
3
E
n,n-I
(l+h) (l+2h)
4
12n
(4.24)
Thus, by (4.22) and (4.24), we have that
E
n,n-u
> 0
< u
n
w
(4.25)
(0,), then (4.20) and (4.25) imply that the terms on the right hand
w
(4.15) are alternate in sign for sufficiently large n uniformly in 2 =< r <= n
Thus, if h
side of
To show that (4.17) is satisfied, we use (4.4) and (4.23) to have from (4.21), (after
521
TRANSLATIVlTY OF ABSOLUTE ROGOSINSKI-BERNSTEIN SUMMABILITY METHODS
straightforward calculations) the result that
3
(u+h- I)
4n
4
3
+
0()
for large n uniformly in 2
-<
u
-<
w
n
(4.26)
This implies that
E
n,n-u-I
u+h
u+h-1
(4.27)
Using (4.20), we have
Bn-u-l,n-r
B
(4.28)
n-u, n-r
Using (4.27) and (4.28), we see that when h 6 (0,1/2), (4.17) is satisfied, and so
(4.16) is satisfied, which by (4.20) and (4.24) shows that the numbers V
are
n,n-r
not bounded for large n uniformly in 2
(4.7) is not
r
--<
w.
n
Hence the necessary condition of
(0,1/2), the transformation given by
absolutely regular, consequently, when h
is not translative
(0,),
Theorem (3.1), fails to hold.
Thus in case h
IBhl
to the left.
As for right translativity, obtain t
n
in terms of
ique to that of left translativlty, we see that when h
and using similar technn+l(0,1/2),
IBhl is not trans-
latlve to the right, and this completes the proof.
REFERENCES
I.
AL-MADI, A.K.
On Absolute Rogoslnski-Bernsteln Trigonometric Summabillty Methods,
(1983), 53-66.
Indian Journal of Mathematics 25
2.
AL-MADI, A.K.
3.
AGNEW, R.P.
4.
PETERSEN, G.M.
445-454.
On Translatlvlty of Rogosinski-Bernsteln Trigonometric Summability
Methods, International Journal of Maths. and Mathematical Sciences (To
Appear).
Rogoslnski-Bernstein Trigonometric Summabillty Methods and Modified
Arithmetic Means, Annals of Maths. 56(3) (1952), 537-559.
A Note on Divergent Series, Canadian Journal of Maths. 4 (1952),