I nternat. J. Math. Math. Sci.
Vol.o 4 No.
(1981) 155-164
155
ON THE CONSISTENCY OF LIMITATION METHODS
FOR (N,Pn) SUMMABLE SEQUENCES
NAND KISHORE
U.K. MISRA
Department. of Mathematics
Berhampur University
Berhampur-760007
Orissa, INDIA
(Received November 8, 1978 and in revised form March 12, 1979)
ABSTRACT.
Two limitation methods, A and B, are said to be consistent for a class
b of sequences, iff, every sequence belonging
that the A-limit equals the B-limit.
to
b is limitable both by A and B and
Any two regular limitation methods are con-
sistent for the class-c of convergent sequences.
However,
this is not true in
general and in fact, corresponding to every bounded non-convergent sequence it is
possible to determine two T-matrices such that they limit the sequence to two
different values.
In this paper, we establish the necessary and sufficient con-
ditions for the consistency of two limitation methods, for
(N,Pn)
summable sequences.
KEY WORDS AND PHRASES. Summable sequence, bmion method, Infinite matrices.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
I.
40A05.
INTRODUCTION.
Let
be an
{Sn
be a sequence of real or complex terms.
infinite matrix over a real or complex field.
Let A
_(am
n
)v,
x
Then the transform, given
by
Tm =n=Zl am nSn
(i.i)
if it exists for every m, is called the A-transform of the sequence {s
lim
m+
plies m
Tm
lim
s, {s n} is said to be A-limitable to s.
Tm
Moreover if
s, the matrix A is said to be regular.
lim
n
}.
s
n n
If
s im-
In 1911, Toeplitz obtain-
N. KISHORE AND U.K. MISRA
156
ed necessary and sufficient conditions for a regular matrix as follows:
A Matrix A
(am n
is regular, iff,
(i)
n
lim
(ii)
m/=
<- M,
mnl
Z
Stoup
0, for every fixed n;
a
mn
n__El= amn Am
(iii)
(1.2)
an absolute constant;
/
I as m
(i 3)
=.
+
(1.4)
Let T be a class of matrices satisfying the conditions (1.2) to (1.4).
matrix of the class
T
is called a Toeplitz matrix or simply a T-matrix.
Any
Thus, a
matrix A is regular if it belongs to the class
Let
{Pn}be
a sequence of constants, real or complex, such that
Pm (Po + Pl +
+
Pm)
# 0, for any
0, i, 2,
m
Then the limitation
method for which
Pm--n
-m
for n
0,
for n > m
<_m
(1.5)
a
m n
is called the
Nrlund method,
or simply (N,p
lar, iff,
m
-
IPnl
(i)
n=o
lim
(ii)
m
/
n)
0(IPnl),
Pm
A
method.
(N,Pn)
method is regu-
(1.6)
for all m,
(1.7)
0.
m
We use the following notations:
(i)
{pn }
(iii)
,
m
1
I
m n=o
tm =p---
(iv)
ZCnx
p(x)
iff,
Po
(1.8)
lPnX
1
(ii)
(v)
n
p(x)
i
Pn
Pm-n Sn’ Pm
n
>0 and
(19)
2
Pn+l
<
PnPn+2
# 0;
(i i0)
(i. II)
Throughout the paper, M is taken for an absolute constant
not necessarily the same at each occurence.
2.
MAIN RESULTS.
Two limitation methods, A and B, are said to be consistent for a class b of
157
CONSISTENCY OF LIMITATION METHODS FOR SEQUENCES
sequences, iff,every sequence belonging to 6 is llmitable both by A and B, and the
Thus, any two matrix methods, generated by the
A-limit is equal to the B-limit.
matrices of the class
T,
are consistent for the class-c of convergent sequences.
However, this is not true in general, and in fact, corresponding to every bounded
non-convergent sequence it is always possible to determine two T-matrices such
[i], page 97).
that they limit the sequence to two different values (see Cooke
A limitation method Q is said to include a limitation method P if every sequence limitable by P is limitable by Q and to the same limit.
dicate this by set theoretic inclusion as, P
Sometimes we in-
Q, meaning thereby that space of
sequences limitable by Q includes that limitable by P.
Two limitation methods, determined by the matrices A
are said to be equivalent for a
lim
m/=
-=
(am n
and B
(Bm n]
class 6 of sequences, iff, for every {s n} E 6
(%_ TmI)
(2.1)
0
where
T
m
__Z
nl
a
s and
mn n
TIm
n
i bmn sn
Hence two limitation methods A and B are consistent for a class 6 of sequences
iff,
(I)
A and B are equivalent, for class 6,
(ii)
A and B limit every sequence
(2.2)
6.
(2.3)
Let 6 be the class of all sequences that are (N,p n) summable.
To ensure condition
(2.2), we prove the following theorems in section 3.
THEOREM I.
Let (N,p n) be a regular Nrlund method and let
{pn }
M.
The
necessary and sufficient conditions that any two limitation methods A and B,
determined respectively by the matrices (am n
such (N,p
(i)
(ii)
(iii)
whe re
n)
and
(bm
n
are equivalent for all
summable sequences are
lim
m/
Ym n
lim
m+=
n
E
Ym n
k--Zo Pk n=Zk m n Cn-k
O, for all fixed n;
(2.5)
0;
<_ M,
(2.4)
for-every m,
.(2.6)
N. KISHORE AND U.K. MISRA
158
(am
(bin
n
n )"
= A and (N,Pn). = B.
(2.3) implies that (N,p n)
Condition
(Ym
n
In section 4, we
prove the following theorem:
the limitation method A, determined by the matrix A
T,
to the class
includes
(N,Pn),
n),
x
E
Then
belonging
iff,
k--Eo Pk n=Ek
m
(am
.
,
{pn}l
Let (N,pn) be a regular Nrlund method and let
THEOREM 2.
am n Cn-k <-
(2.7)
M,
where ck is as defined in (1.9).
We required the followng lemma of Kaluza (see Hardy [’2 ], page 68) in proving
our theorem.
LEMMA.
If p(x)
EPnxn
is convergent for
p(x) -i
Ix
+ ClX +
i
<
I, and
{pn}l
E
,
and further
c2x2+
then
E[Cnl
3.
<_ 2.
Epn--oo,
If
n=El fen
then
2.
PROOF OF THEOREM i.
At the outset we observe that if (N,p n) is regular and
then in view of the regularity condition,
E
Pnxn
is absolutely convergent for
n=o
is also
Ix
so, for
the series
n
r.PnX
< i.
Ix[
We have
Pn
n
n
n--o
for each n,
i and the series
(3.1)
x),
xl
< I.
n
P
-n-
Sk)
n
x
n
lPnX
k__Eo Pn_kSk
n
(k__Eo
Pn >-- O,
< i, as such the series
P xn (I
i
n
< i, we have
n
E t P x
n--o n n
Hence
n/
is absolutely convergent for
t
Then, for
Pn+l
But then the series (3.1) equals
Now we lay down the proof.
(If part):
xl
llm
p(x) s(x)
and accordlngly
CONSISTENCY OF LIMITATION METHODS FOR SEQUENCES
n=o tnxn
i
s(x)
p(x)
n
n--oF"
Equating the coefficients of x
n,
159
(kEo= tkPkCn_ k)
x
n
from both sides, we have
k__Zo tkPkCn_ k"
sn
(3.2)
Now
n
Z
n--o
Let us put t k
Z
n=o
t
Ym n
Ym n
s
+ k’
s
n--o
lim
where t
t
n
Ym n k-Z- PktkCn-k
k=Zo tkPk n= Ym n Cn-k
n
k-/oo t k and
Pk n Ym n Cn-k
k
+
{ k}
is a null sequence
k
k
n=
Then
Ym nCn-k )I 6
n
Taking the limit as m
t
kZo Ym n k--Zo PkCn-k + k--Zo (Pk n--Zk Ym n Cn-k) k
t
k=Zo m n + k--Zo (Pk n-- m n Cn-k] k"
and making use of condition (2.5), we have
/
lira
n--o m
__o (Pk n--k m nCn-k
lim
Sn
n
m
k
k,
=o
/
k
say,
whe re
dm k
Pk
nk= Ym
n
Cn-k
Hence the proof is completed if we show that
llm Z
But since {
k}
/
0
as k
/
=o,
k
(ii)
O.
it is sufficient to show
Z
(i)
dmk
-
lira
m
Here (3.3) follows from (2.6).
dm kl
< M, for every m;
d
m k-- 0, for every fixed k.
(3.3)
(3.4)
For establishing (3.4), it is sufficient to show
that
lim
for every fixed k.
(nk
m nCn-k
0,
(3.5)
160
N. KISHORE AND U.K. MISRA
Cn]
Now since Z
is convergent, for any arbitrary small
> 0 we can have n
o
such that
l
n>no
]cnl
<
2-
(3.6)
ICnl
<A,
(3.7)
where M is as specified in (2.6).
Further let
Z
n<no
where A is a finite constant.
In view of (2.4) and the fact that n o is a finite
positive integer, we can have m dependent
o
IYm nl
for all m > m
o
)
<
and all nffik, k+l,
on
(3.8)
2n
o
ko
n=Zh Ym n n
h
]Ym h
such that
+ k.
Also
PkCn-h-k
=I ko Pk nffi+k Tm n=n-h-kl
<
k=Zo Pk+h n=+k Wm nCn-h-k
<
vZh Pv inv<fm ncn-v
<
(3.9)
M, by (2.6),
Hence, finally
for all m and h.
nZkY
m
lnr.=o m,n+k=n
nCn_k
n
r..
<
n=o
n
<
IA
m, n+kCn
o
__Z
n-o
ICn]
<
Hence
mli_oo
d
m k
+
+M Z
n>n o
.
12--
A + M
>-:.
nno "Y’m,n+kCn[
[On[ for m > m,o
’2’
by (3.8) and (3.9).
by (3.7) and (3.6).
(3.10)
0, for every fixed k, and the "if part" of the theorem is
proved.
(Only
if
part):
Let the limitation method of A and B be equivalent for all (N,p
n)
summable
CONSISTENCY OF LIMITATION METHODS FOR SEQUENCES
sequences.
161
Then
lira
m -o
where (a
(bm n)
m n
(Ym
n
nE Ymn sn
0,
)’ and {sn}
is a
(N,pn) summable sequence.
We have
7.
n--o
s
ko (Pk n--Ek Ym nCn-k tk"
Ymnn
As {s } is a
n
(i)
take
..(N,Pn summable sequence
-K-n
{h}
{s }
so that t
k
n
n
P
for every k > h
k
Then
EYmn sn k=Eo Pk-h nk Ym nCn-k
n
n
n=ol Ym n k--Zo Pk-hCn-k
n-h
n=o
Ym n ko Pk Cn-k
Hence
lm
m
Zn m n sn
li+mo
m
0, for every fixed h.
Ym n
(3.11)
Thus hypothesis (2.4) is necessary.
(ii) Take s n
I in
lm
m
{Sn}
Z
n
so that t
Ym n n
k
I, for every k.
Then
m
m
l+Im
n
no Ym
n
ko
,%
n
--o,
m/oono
PkCn-k
(3.12)
which is hypothesis (2.5).
(iii) Take tk
Y.
lim
m/oo
n
t
s
-mnn
+Ek, wherel{Ek }
lira
m
t
/
is a null sequence.
ko Pk nk Ym nCn-k +
k--
n=
nCn-k
m -+
Then
k--
n=
k’ by (3.12).
N. KISHORE AND U.K. MISRA
162
But
lira
m
/
7
n
m ns --0,
lim
m
for all null
sequenceS{Sn }
for all
n
ko (Pk n--k m nCn-k k
/
(N,Pn)
that are
summable, as such
0
sequence{6k}.
Hence
Sup 7.
k
m
Pkl nk m nCn-kl <- M
Thus Theorem i is established.
4.
PROOF OF THEOREM 2.
(If part):
We have
Tm
7. a
s
n=o m n n
n
a
n=o m n k o
=E
Y.
n=Ek
k__Eo tkPk
tkPKCn_ k
a
m
nCn-k
by (3 2),
ko bm k tk,
say,
where
b
am nCn-k.
Pk n--Ek
m k
(4. I)
In order to establish this part, it is sufficient to show that (bm k
that is, it belongs to
is regular,
T.
Clearly
Ibm k
Since
Pk
In=k
a
m
< M by (2 7)
nCn-k
(am n T, for ewery fixed positive integer n and
’mla n
<
.,
for all m >
mo
>
O,
(n, 6),
(4.2)
and also
Sup
m,n
am,n
< M.
(4.3)
Now, making use of (4.2) and (4.3), and proceeding along the lines of (3.10), we
can easily establish that
lim b
m /oo m k
lim
m /
Pk
n k
a
c
m n n-k
Finally, since
n
k
PkCn_k
I, for every n,
0, for every fixed k.
163
CONSISTENCY OF LIMITATION METHODS FOR SEQUENCES
n
ko bm k =’k=Zo Pk
n=
7. a
n=o m n
Hence (b
m k
a
m
nCn-k
A
/
Z
n=o
a
m n
k
PkCn_k
i, as m-+ oo.
m
satisfies all the regularity conditions
This proves the "if
part"
of Theorem 2.
(Only
if
part):
We have
Tm
Z
a
s
m n n
Z
a
m n
n=o
n=o
Since A includes
(N,Pn)
(b
k__ tkPkCn_k
-k--Zo tkPk k
a
ko bm k tk,
say.
m k
is regular
ko Ibm kl k--EL Pk
c
mnn-k
Thus
Ink am nCn-kl
< M,
which is the required condition.
This completes the proof of Theorem 2.
5
CLOSING REMARKS.
Theorem i generalizes the result of Zaman [3].
form if A and B belong to the class
THEOREM 3
Let
(,pn)
T
It assumes a much simplified
and we have:
be a regular NSrlund method and let
{Pn ) 6 4.
Then a
he-
cessary and sufficient condition that any two limitation methods, determined by
the matrices A
for all such (N,p
(am n
n
(b
and B
m n
T,
belonging to the class
are equivalent
summable sequences, is that (2.6) hold.
Theorem 2 and 3 together lead to the following Theorem of consistency of
matrix limitation methods for (N,p
THEOREM 4.
Let
(N,pn)
n)
summable sequences.
be a regular N6"rlund method and let
{pn } 6.
Then
the necessary and sufficient conditions that any two limitation methods, determined
by the matrices of the class
are that they include
ACKNOWLEDGEMENT.
T,
are consistent for all
(N,Pn)
summable sequences
(N,Pn).
Authors are indebted to the referee for his valuable criticism
and useful suggestions on the original manuscript.
164
N. KISHORE AND U.K. MISRA
REFERENCES
i.
Cooke, R.G. Infinite Matrices and sequence spaces, Macmllan, 1950.
2.
Hardy, G.H.
Divergent Series, Clarendon Press, Oxford, 1949.
3.
Zaman, M.B.
On absolute equivalence of T-matrlces for (C,r) summable sequences,
Jour. Indian Math. Soc. 36 (1972), 89-96.