Matrix Transforms into a Set of
-absolutely
A
-summable Sequences
ANTS AASMA1, PINNANGUDI N. NATARAJAN2
1Department of Economics and Finance,
Tallinn Univ. of Technology,
Akadeemia tee 3-456, 12618,
ESTONIA
2Old no. 2/3, new no.3/3 Second main road, R. A. Puram, Chennai 600028,
INDIA
Abstract: - Suppose A is a matrix having real or complex entries and
- a monotonically increasing strictly
positive sequence, i.e., the speed. In this paper, the notions of
-reversibility of A,
A
-boundedness, and
A
-
summability of sequences are recalled, and the notion of
-absolute
A
-summability of sequences is
introduced. Also, there are characterized matrix transforms from the set of all
A
-bounded, or the set of all
A
-
summable, or the set of all 1- absolute
A
-summable sequences into the set of all
-absolutely (
1
)
B
-
summable sequences for a normal or
-reversible matrix A and a matrix
()
nk
Bb
with
0
nk
b
,
kn
, and
for another speed
.
Key-Words: - Matrix transforms,
-reversibility of matrices, boundedness with speed, convergence with speed,
zero-convergence with speed, summability with speed,
-absolute summability with speed.
1 Introduction
First, we note that the present paper is the
continuation of [1]. Therefore, we strictly follow the
notations and concepts of [1].
The speed of convergence of sequences can be
treated in different ways, see, for example, [2], [3],
[4], [5], [6], [7], [8]. We use, as in [1], the tools,
introduced in [2] and [7].
However, in general, we do not expect the
boundedness of
, the most important case is
(1),
kO
because in this case relation
lim :
k
k
xs
and
: ( )
k k k
v x s
(1)
for a convergent sequence
()
k
xx
allows to
evaluate the quality of convergence of converging
sequences. Indeed, let
1
x
and
2
x
be two convergent
sequences with the finite limit s. If
()
k
vc
(or
(1)
k
vO
) for
1
xx
, and
()
k
vc
(or
(1)
k
vO
)` for
2
xx
, then the sequence
1
x
converges "better"
(more precisely, faster) than sequence
2
x
. Thus
,
in the case
(1),
kO
measures the speed of
approaching to the limit s for the observed
sequences.
Let
()
nk
Aa
be an arbitrary matrix with real or
complex entries. Following [6] (see also [2]), a
sequence
()
k
xx
is called
A
-bounded (
A
-
summable), if
Ax l
(correspondingly
Ax c
).
Definition 1. If
,Ax l
then we tell that a
convergent sequence
()
k
xx
is
-absolutely
A
-
summable,
Let
A
l
- the set of
A
-bounded sequences,
A
c
- the set of all
A
-summable sequences, and
A
l
the set of all
-absolutely
A
-summable
sequences. Let
00
: : ,
A
A
c x c Ax c
: : .
A
c x Ax c
It is easy to see that
0,
AA
A A A
l c c l c
and
0AA
AA
l c c c
a bounded
.
Matrix transformations, boundedness and
convergence with speed are widely used in
approximation theory to transform non-convergent
sequences into convergent ones or to transform
Received: June 12, 2024. Revised: October 23, 2024. Accepted: November 13, 2024. Published: December 15, 2024.
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Ants Aasma, Pinnangudi N. Natarajan
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891
Volume 23, 2024
convergent sequences into “better” convergent
sequences, [6], [9], [10], [11].
In general, the problems of improvement of the
quality of convergence of sequences by matrix
transformations are considered in many works, [1],
[12], [13], [14], [15], [16]. Moreover, the
applications in theoretical physics can be found, for
example, in [17] and [18].
Let further
: ( )
k
be another speed,
()
nk
Bb
a lower triangular infinite matrix, and a matrix
()
nk
Aa
is normal or
-reversible. We recall that A
is normal, if
0
nk
a
for
,kn
and
0
nn
a
for each
n, and
-reversible, if the system of equations
nn
z A x
has a unique solution for each sequence
.
n
zc
Matrix transforms from
,XY
where X is
one of the sets
,c
0
c
,
l
,
1
l
,
,
A
c
0A
c
,
A
l
or
1A
l
, and Y one of the sets
,c
0
c
,
l
,
l
(
1
),
,
B
c
0B
c
,
B
l
or
1B
l
have been studied in [6],
and in several works of the authors of the current
paper, we mention only [1] and [2].
In this paper, we characterize the matrix
transforms from
,
A
c
0A
c
,
A
l
and
1A
l
into
B
l
(
1
).
2 Auxiliary Results
In this section, we present some results, which we
use in next sections in the proofs of the main
theorems of the present paper.
The following Lemmas 1 – 3 have been proved
in [19] and [20].
Lemma 1. For
0
( ) ( , )
nk
A a c c
it is necessary and
sufficient that
lim :
nk k
m
aa
(finite) (2)
sup .
nk
nk
a
(3)
Moreover,
lim n k k
nk
A x a x
(4)
for every
0
( ) .
k
x x c
Lemma 2. For
( ) ( , )
nk
A a c c
it is necessary and
sufficient that conditions (2), (3) hold, and
such that
lim : .
nk
nk
a
(5)
Moreover, if
lim k
k
xs
for
()
k
x x c
, then
lim ( ) .
n k k
nk
A x s x s a
Lemma 3. Next assertions are equivalent:
(a)
( ) ( , )
nk
A a l c
.
(b) Relations (2), (3) hold and
lim 0.
nk k
nk
aa
(6)
(c) Relation (2) is satisfied and
nk
k
a
uniformly with respect to n. (7)
In addition, if one of the assertions (a)-(c) holds,
then (4) is satisfied for each
( ) .
k
x x l
The following Lemmas 4 – 6 have been proved
in [20].
Lemma 4. For
()
nk
Aa
1
( , )lc
it is necessary and
sufficient that (2) holds, and
,
sup .
nk
nk
a
(8)
In addition, relation (4) is valid for each
1
( ) .
k
x x l
Lemma 5. For
()
nk
Aa
1
( , )ll
(
1
) it is
necessary and sufficient that
sup .
nk
kn
a
Lemma 6. For
()
nk
Aa
0
( , ) ( , )l l c l
(
1
) it
is necessary and sufficient that
sup nk
Kn k K
a
for all finite subsets K from N: =
0,1,2,... ,
or
*
nk
n k K
a
for every
* N.K
.
Let
()
nk
Aa
be a
-reversible matrix. For
,
A
xc
let
: lim , : ( ), : lim ,
n n n n n
nn
A x d A x d d
and
: ( ),
k
: ( )
k
,
()
kj
(for every j) satisfy the
system
y Ax
respectively to
( ),
nn
y
( / )
nn n
y
: ( )
j
nj
yy
(where
1
nj
if
nj
and
0
nj
if
nj
).
Lemma 7 ([2], Corollary 9.1). For a
-reversible
( ),
nk
Aa
each member
k
x
of a sequence
()
kA
x x c
can be represented in the form
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Volume 23, 2024
( ), .
kn kn
k k k n
nn
nn
x d d d
(9)
Remark 1. If
0
( ) ,
kA
x x c
then
0.d
Hence for
every member
k
x
of
0
()
kA
x x c
we may write
( ), .
kn kn
k k n
nn
nn
x d d
(10)
3 The Set
,
AB
ll
,
1
Let throughout in this section
()
nk
Aa
be a normal
matrix with its inverse matrix
1: ( )
nk
A
,
()
nk
Bb
a triangular matrix, and
()
nk
Mm
an arbitrary
matrix
In the beginning, we find necessary and
sufficient conditions for the existence of
Mx
on
.
A
l
In this case for all
: ( )
kA
x x l
we can
write
0 0 0 0
j j j
k
n
nk k nk kl l jl l
k k l l
m x m y h y
:
ll
y A x
,
where
:
nn
jl
Hh
is the lower triangular matrix for
each n, defined by
:,
j
n
jl nk kl
kl
hm
.lj
Consequently, for the existence of
Mx
for each
A
xl
it is necessary and sufficient that
( , )
n
H l c
for all n. Thus, we obtain ([2],
Proposition 8.1)
Proposition 1. For the existence of
Mx
on
A
l
it
is necessary and sufficient that for every n,
lim :
n
jl nl
j
hh
(finite), (11)
0
lim
j
n
jl
jl
h
(finite) (12)
sup ,
n
jl
jll
h
(13)
lim 0.
n
jl nl
jll
hh
(14)
Moreover, instead of (13) it is possible to set
nl
ll
h
(15)
for all n
Remark 2. With the help of Lemma 3 c) it is
possible to prove that instead of (13) and (14) we
can put
n
jl
ll
h
uniformly in j for all n. (16)
Next, we need the matrix
( ) ;
nk
G g BM
i.e.,
0
:,
n
nk nl lk
l
g b m
and lower triangular matrices
:,
nj
nl
where
:,
j
j
nl nk kl
kl
g
.lj
Theorem 1. Let
(1)
nO
and
1.
For
,
AB
M l l
it is necessary and sufficient that
(11) – (14) hold,
lim :
nl l
j
(finite), (17)
lim 0,
nl l
nll
(18)
sup ,
nl
nll
(19)
sup nl l
n
Kn l K l
(20)
for every finite
N,K
where
: lim j
nl nl
j
, (21)
and
( ) ,
nl
0
: lim .
j
j
n nl
jl
(22)
Moreover, instead of (19) it is possible to set
.
l
ll
(23)
Proof. Necessity. Suppose that
,.
AB
M l l
In this case
Mx
exists on
.
A
l
This implies that
(11) - (14) hold by Proposition 1, and
nn
B y G x
(24)
on
,
A
l
since B is lower triangular.
Hence
,
A
G l l
by (24) .
Moreover,
00
jj
j
nk k nl l
kl
g x A x
(25)
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for all
.
A
xl
As A is normal, then we can find
A
xl
satisfying the equality
.
l
A x e
Consequently (22) holds by (25).
As A is normal, then for every bounded
sequence
()
n
we can find
A
xl
satisfying the
relations
lim :
n
n
Ax
and
( ).
n n n
Ax
(26)
As from (26) we have
,
n
n
n
Ax
then, using (25) and (26), we obtain for all
A
xl
that
0 0 0
.
j
j j j
jnl
nk k nl l
k l l l
gx
(27)
Since
n
Gx
converges on
,
A
xl
and there
exist the finite limits
n
by (22), then
: / ( , )
nj
nl l lc
for all n. Consequently, using
Lemma 3, we conclude from (27) that
nl
n n l
ll
Gx
(28)
for every
.
A
xl
We note that the existence of
the finite limits
nl
(see (21)) follows from the
existence of the transform
Mx
on
.
A
l
Moreover,
the finite limit
lim :
n
n
exists by (22). Hence
from (28), we can conclude that the matrix
: / ( , ).
nl l lc
This implies that conditions
(17) – (19) hold and
lim l
nl
nll
Gx
(29)
for every
A
xl
by Lemma 3. Writing
( lim ) ( )
n n n n n
n
nl l
nl
ll
G x G x
(30)
for every
A
xl
, we obtain with the help of (22)
that
,: ( ) / ( , ).
n nl l l ll
Therefore
condition (20) is satisfied by Lemma 6.
Finally, (23) holds by (18) and (19).
Sufficiency. Let relations (11) – (14) and (17) – (22)
be fulfilled. In this case
Mx
exists on
A
l
by
Proposition 1, and (24) – (27) are satisfied on
A
l
.
Similarly to the necessity part, with the help of (22)
and Lemma 3, it is possible to show that (27)
implies (28) for all
A
xl
. As (22) holds and
( , )lc
by Lemma 3 (due to the validity of (17)
– (19)), then from (28) we can conclude (again by
Lemma 3) that equation (29) is valid for every
A
xl
. Hence relation (30) also is satisfied on
A
l
, and
,( , )ll
by Lemma 6. Therefore
,
AB
M l l
due to (22).
Finally, instead of (19) we can put (23) since
(19) holds by (18) and (23).
If
sup ,
n
n
then
lim 0
n
n
in (26). Therefore,
in this case
0
( , )
ncc
for every n. Thus, applying
Lemma 1, we can immediately to formulate
Theorem 2. Let
(1)
nO
and
1.
For
,
AB
M l l
it is necessary and sufficient that
(11) – (14), (17) and (19), (20), (22) hold.
Remark 3. With the help of Lemma 3 c) we obtain
that in Theorem 1, instead of (13), (14) we may set
(16), and instead of (18) and (19) – the relation:
nl
ll
uniformly with respect to n.
Remark 4. It is possible to show with the help of
Lemma 6 that instead of (20) in Theorems 1 and 2
we can put:
*
nl l
n
n l K l
(
*NK
). (31)
4 The Sets
,
AB
cl
,
0,
AB
cl
and
1,,
AB
ll
1
Let throughout in this section
()
nk
Aa
be a
-
reversible matrix,
()
nk
Bb
a triangular matrix, and
()
nk
Mm
an arbitrary matrix
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Proposition 2 ([2], Proposition 9.5). For the
existence of
Mx
on
A
c
it is necessary and sufficient
that (11), (13) hold, and for every n,
nk k
k
m
, (32)
nk k
k
m
.
(33)
Proposition 3. For the existence of
Mx
on
0
()
A
c
it
is necessary and sufficient that (11), (13) and (32)
hold.
Proposition 4. For the existence of
Mx
on
1
()
A
l
it is
necessary and sufficient that (11), (32) hold, and
(1)
n
jl
n
l
hO
. (34)
Propositions 3 and 4 can be proved in a similar
fashion to Proposition 2 (presented in [2]);
therefore, we omit the proofs of these propositions.
Further, we apply matrices
:
nj
nl
, where
:.
j
j
nl nk kl
kl
g
If
Mx
exists on
A
c
(on
0
( ) ,
A
c
or on
1
()
A
l
), then
finite limits (21) exist.
Theorem 3. Let
1.
For
,
AB
M c l
it is
necessary and sufficient that (11), (13), (17), (19),
(20), (32), (33) are satisfied, and
,.
G
l
(35)
Proof. Necessity. Suppose that
,
AB
M c l
. In
this case
Mx
exists on
.
A
c
This implies that (11),
(13), (32), and (33) are satisfied by Proposition 2,
and relation (24) is valid on
A
c
since B is lower
triangular. Consequently
,
AG
cl
and
n
Gx
converges on
.
A
c
This implies that condition (35)
holds due to
,.
A
c
As each member
k
x
of
: ( )
kA
x x c
may be presented in the form (9) by
Lemma 7, then, we have:
0 0 0
()
j
j j j
nl
nk k nk k nk k l
k k k l l
g x g d g d d
(36)
for all
.
A
xc
As
n
Gx
converges on
,
A
c
then from
(36) it follows, by (35), that the matrix
n
for each
n transforms this sequence
0
()
l
d d c
into c. We
show that
n
transforms every sequence
0
()
l
dc
into c. Indeed, for every sequence
0
()
l
dc
there
exists a sequence
()
l
dc
with
,
ll
d d d
where
lim .
l
l
dd
Then
0
/.
ll
d d c
Moreover, for
this sequence, we can find
()
l
z z c
so that
/,
l l l
d d z
: lim ,
l
l
z
As A is
-reversible, then for each
()
l
z z c
with
: lim ,
l
l
z
there exists a convergent sequence x
with
.
ll
z A x
So we showed that, for each
0
()
l
dc
we can find
A
xc
so that
( ).
l l l
d A x
Hence
0
( , ).
ncc
Therefore (36) implies, by Lemma 1,
that:
()
nl
n n n l
ll
G x G dG d d
(37)
for all
.
A
xc
From (37) we conclude by (35) that
0
( , ).cc
This implies by Lemma 1 that
conditions (17), (19) hold and:
lim lim lim ( )
l
n n n l
n n n ll
G x G d G d d
(38)
for every
.
A
xc
Consequently we can write:
lim lim
lim ( )
n n n n n n
nn
nl l
n n n n l
nll
G x G x G G
d G G d d
(39)
for each
.
A
xc
From (39) it follows, by (35), that
,0
( , ).cl
Thus (20) holds by Lemma 6.
Sufficiency. Let relations (11), (13), (17), (19),
(20), (32), (33) and (35) be fulfilled. In this case
Mx
exists on
A
c
by Proposition 2. Hence (24),
(36) are satisfied for every
A
xc
and
0
( , )
ncc
for all n. Then it is possible to take the limit under
the summation sign in the third summand of (36) by
Lemma 1. Thus, using (35), from (36) we obtain
that (37) holds on
A
c
by Lemma 1. Moreover,
0
( , )cc
by (17) and (19). Hence from (37) we
have, by (35) and Lemma 1, that (38) holds for
every
.
A
xc
Then relation (39) also is satisfied for
every
A
xc
, and
,0
( , )cl
by Lemma 6
because condition (20) holds. Therefore
,
AB
M c l
by (35).
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If instead of
A
c
to take
0A
c
in Theorem 3,
then
0d
in the proof of this theorem, and instead
of Proposition 2 it is necessary to use Proposition 3.
Therefore, we can immediately to formulate the
following result.
Theorem 4. Let
1.
For
0,
AB
M c l
it is
necessary and sufficient that (11), (13), (17), (19),
(20), (32) are satisfied, and
.
G
l
Remark 5. As in Section 3. it is possible to show by
Lemma 6 that condition (20) in Theorems 3 and 4
can be replaced by condition (31).
Theorem 5. Let
1.
For
1,
AB
M l l
it is
necessary and sufficient that (11), (17), (32), (34)
are satisfied,
1,
G
l
and
,
sup ,
nl
nl l
1
sup .
n nl l
nn
l
5 Conclusion
This paper is the continuation [1]. Let
1,
and
,
be speeds of convergence; i.e., monotonically
increasing strictly positive sequences In the current
paper, we characterized the matrix transforms from
the λ-boundedness domain of a normal matrix A
(with real or complex entries), and from the λ-
convergence or from the absolute λ-convergence
domain of a λ-reversible matrix A into the
-
absolute
-convergence domain of a triangular
matrix B (with real or complex entries), .
Further, we intend to generalize the results
of this paper to abstract structures. For example, we
will study matrix transforms over ultrametric
spaces. Also, we try to apply our results in
approximation theory, for example, for the
estimation of approximation orders of Fourier
expansions.
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-absolutely
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