Matrix transforms into the set of α-absolutely convergent sequences

with speed and the regularity of matrices on the sub-spaces of c

ANTS AASMA1, PINNANGUDI N. NATARAJAN2

1Dept. of Econ. and Fin., 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: - Let α > 1.The α-absolute convergence with speed, where the speed is defined by a monotonically

increasing positive sequence µ, has been studied in the present paper. Let lµ

αbe the set of all α-absolutely µ-

convergent sequences and Xa sequence space defined by another speed λ. Necessary and sufficient conditions

for a matrix A(with real or complex entries) to map Xinto lµ

αhave been presented. It is proved as an example

that the Zweier matrix Z1/2 satisfies these necessary and sufficient conditions for certain speeds λand µ. The

notion of regularity on the subspace Xof the set cof converging sequences is defined, and also, necessary and

sufficient conditions for a matrix Ato be regular on certain X⊂care presented. It has also been shown that there

exists an irregular matrix, which is regular on the subspace Xof c.

Key-Words: Matrix transforms, boundedness with speed, convergence with speed, α-absolute convergence with

speed, Zweier matrix, regularity of matrices, regularity of a matrix on the subset of c.

Received: March 12, 2023. Revised: October 11, 2023. Accepted: November 10, 2023. Published: January 26, 2024.

1 Introduction

Let X, Y be two sequence spaces and A= (ank)

be an arbitrary matrix with real or complex entries.

Throughout this paper we assume that indices and

summation indices run from 0 to ∞unless otherwise

specified. If for each x= (xk)∈Xthe series

Anx:= X

ankxk

converge and the sequence Ax = (Anx)belongs to

Y, we say that Atransforms Xinto Y. By (X, Y )we

denote the set of all matrices, which transform Xinto

Y. Let ωbe the set of all real or complex valued se-

quences. Further we need the following well-known

sub-spaces of ω:c- the space of all convergent se-

quences, c0- the space of all sequences converging to

zero, l∞- the space of all bounded sequences, and

lα:= {x= (xn) : X

|xn|α<∞}, α > 0.

For estimation and comparison of speeds of conver-

gence of sequences are used different methods, see,

for example, [1], [2], [3], [4], [5], [6], [7], [8]. We

use the method, introduced in [6] and [7] (see also

[1]). Let λ:= (λk)be a positive (i.e.; λk>0for

every k) monotonically increasing sequence. Follow-

ing [6] and [7] (see also [1]), a convergent sequence

x= (xk)with

lim

kxk:= sand vk=λk(xk−s)(1)

is called bounded with the speed λ(shortly, λ-

bounded) if vk=O(1) (or (vk)∈l∞), and conver-

gent with the speed λ(shortly, λ-convergent) if the

finite limit

lim

kvk:= b

exists (or (vk)∈c). In the following we define the

notion of α-absolute convergence with speed.

Definition 1. We say that a convergent sequence

x= (xk)with the finite limit sis called α-absolutely

convergent with speed λ(or shortly, α-absolutely λ-

convergent), if (vk)∈lα.For α= 1 a sequence x

is said to be absolutely convergent with the speed λ

(shortly, absolutely λ-convergent).

We denote the set of all λ-bounded sequences by

lλ

∞, the set of all λ-convergent sequences by cλ, and

the set of all α-absolutely λ-convergent sequences by

lλ

α. Moreover, let

cλ

0:= {x= (xk) : x∈cλand lim

kλk(xk−s) = 0}

and

lλ

∞,0={x= (xk) : x∈lλ

∞∩c0}.

It is not difficult to see that

lλ

α⊂cλ

0⊂cλ⊂lλ

∞⊂c, lλ

∞,0⊂lλ

∞⊂c.

In addition to it, for unbounded sequence λthese in-

clusions are strict. For λk=O(1), we get cλ=lλ

∞=

Let e= (1,1, ...),ek= (0, ..., 0,1,0, ...), where 1

is in the k-th position, and λ−1= (1/λk). We note

that

e, ek, λ−1∈cλ;e, ek∈lλ

α.

A matrix Ais said to be regular if A∈(c, c)and

limnAnx=limnxnfor every sequence x= (xn)∈

Definition 2. Let Xbe a subspace of c; i.e., X⊆c.

We say that a matrix Ais regular on X,if limnAnx=

limnxnfor every sequence x= (xn)∈X.

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Let µ:= (µn)be another speed of con-

vergence; i.e., a monotonically increasing positive

sequence. Matrix transforms between the sub-

sets of cdefined by the speeds λand µhave

been studied by the authors of the present work

in several papers. For example, in [9] the sets

lλ

∞, cµ,lλ

∞, lµ

∞,0,lλ

∞, cµ

0,cλ, lµ

∞,0,cλ, cµ

0,

lλ

∞,0, lµ

∞,lλ

∞,0, lµ

∞,0,lλ

∞,0, cµ,lλ

∞,0, cµ

0,

cλ

0, lµ

∞,cλ

0, lµ

∞,0,cλ

0, cµand cλ

0, cµ

0have

been characterized. A short overview on the conver-

gence with speed has been presented in [1].

The boundedness and convergence with speed are

tightly connected with the problems of convergence

acceleration and improvement by matrices. These

problems have been studied by one author of the

present paper (see, for example, [1]), and by sev-

eral other authors; for example, [10], [11], [12], [13],

[14], [15], [16] and [17]. Moreover, in [16] and

[17], the λ-convergence and the λ-boundedness in

abstract spaces, considering instead of a matrix with

real or complex entries a matrix, whose elements are

bounded linear operators from a Banach space Xinto

a Banach space Y, have been studied.

We note that the results connected with con-

vergence, absolute convergence, α-absolute λ-

convergence and boundedness with speed can be

used in several applications. For example, in the

theoretical physics such results can be used for

accelerating the slowly convergent processes, a good

overview of such investigations can be found, for ex-

ample, from the sources [18] and [19]. These results

also have several applications in the approximation

theory. Besides, in [1] such results are used for the

estimation of the order of approximation of Fourier

expansions in Banach spaces.

In the present paper we describe the matrix trans-

forms related to the α-absolute λ-convergence for the

case α > 1, giving the characterization for the sets

(lλ

∞, lµ

α),(cλ, lµ

α),(cλ

0, lµ

α),(lλ

1, lµ

α), and necessary and

sufficient conditions for the regularity of a matrix A

on lλ

∞,cλand cλ

0. Also we will present an example of

irregular matrix, which is regular on cλ

0and on cλ, but

not on lλ

∞for some λ. Moreover, we will prove that

this irregular matrix is regular on cλ

0, on cλand on lλ

∞

for another speed λ.

2 Auxiliary results

For the proof of main results we need some auxiliary

results.

Lemma 1 ([20], p. 44, see also [21], Proposition 12).

A matrix A= (ank)∈(c0, c)if and only if

there exists limits lim

nank := ak,(2)

|ank|=O(1) .(3)

Moreover,

lim

nAnx=X

akxk(4)

for every x= (xk)∈c0.

Lemma 2 ([20], p. 46-47, see also [21], Proposition

11 or [22], p. 17-19). A matrix A= (ank)∈(c, c)if

and only if conditions (2), (3) are satisfied and

there exists τ with lim

ank := τ. (5)

Moreover, if limkxk=sfor x= (xk)∈c,then

lim

nAnx=sτ +X

(xk−s)ak.

A matrix Ais regular if and only if conditions (2), (3)

and (5) are satisfied with ak= 0 and τ= 1.

Lemma 3 ([20], p. 51, see also [21], Proposition 10).

The following statements are equivalent:

(a) A= (ank)∈(l∞, c).

(b) The conditions (2), (3) are satisfied and

lim

|ank −ak|= 0.(6)

|ank|converges uniformly in n. (7)

Moreover, if one of statements (a)-(c) is satisfied, then

the equation (4) holds for every x= (xk)∈l∞.

Lemma 4 ([21], Proposition 17 or [22], pp. 25-26).

A matrix A= (ank)∈(l1, c)if and only if condition

(2) is satisfied and

ank =O(1) .(8)

Moreover, the equation (4) holds for every x=

(xk)∈l1.

Lemma 5 ([21], Proposition 21). A matrix A=

(ank)∈(l∞, c0)if and only if

lim

|ank|= 0.

Lemma 6 ([21], Proposition 22). A matrix A=

(ank)∈(c, c0)if and only if conditions (2) and (5)

with ak= 0,τ= 0,and condition (3) are satisfied.

Lemma 7 ([21], Proposition 23). A matrix A=

(ank)∈(c0, c0)if and only if condition (2) with

ak= 0,and condition (3) are satisfied.

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Lemma 8 ([21], Proposition 68). A matrix A=

(ank)∈(l1, lα)for α > 1if and only if

|ank|α=O(1) .

Lemma 9 ([21], Proposition 63). A matrix A=

(ank)∈(l∞, lα) = (c, lα) = (c0, lα)for α > 1if

and only if

nX

k∈K

ank

=O(1)

for every finite subset Kof N:= {0,1,2, ...},or the

series

nX

k∈K

ank

is convergent for arbitrary subset K∗of N.

3 Main results

3.1 Matrix transforms into the set lµ

First we prove

Theorem 1. Let λn=O(1).A matrix A= (ank)∈

lλ

∞, lµ

αfor α > 1if and only if condition (2) is sat-

isfied, and

Ae = (τn)∈lµ

α, τn:= Ane=X

ank,(9)

|ank|

λk

=O(1),(10)

lim

|ank −ak|

λk

= 0,(11)

µα

n

X

k∈K

ank −ak

λk



=O(1),(12)

where Kis an arbitrary finite subset of N.

Proof. Necessity. Assume that A∈lλ

∞, lµ

α. As

e∈lλ

∞and ek∈lλ

∞, then conditions (2) and (9) hold.

Since, from (1) we have

xk=vk

λk

+s;s:= lim

kxk,(vk)∈lα

for every x:= (xk)∈lλ

∞, it follows that

Anx=X

ank

λk

vk+sτn.(13)

As (τn)∈lµ

αby (9), then, from (13) we obtain that

the matrix

Aλ:= ank

λk

transforms this sequence (vk)∈l∞into c. In ad-

dition, for every sequence (vk)∈l∞, the sequence

(vk/λk)∈c0.But, for (vk/λk), there exists a conver-

gent sequence x:= (xk)with s:= limkxk, such that

vk/λk=xk−s. So we have proved that, for every se-

quence (vk)∈l∞there exists a sequence (xk)∈lλ

∞

such that vk=λk(xk−s). Hence Aλ∈(l∞, c).

This implies, by Lemma 3 ((a) and (b)), that condi-

tions (10) and (11) are satisfied, since for Aλcondi-

tions (3) and (6) take correspondingly the forms (10)

and (11), and the finite limit

ϕ:= lim

nAnx=X

λk

vk+slim

nτn

exists for every x∈lλ

∞. Writing

µn(Anx−ϕ) = µnX

ank −ak

λk

+sµn(τn−lim

nτn),(14)

we obtain, by (9), that the matrix Aλ,µ ∈(l∞, lα),

where

Aλ,µ := µn

ank −ak

λk.

Hence condition (12) is satisfied by Lemma 9, since

for Aλ,µ ∈(l∞, lα)the first condition of Lemma 9

takes the form (12).

Sufficiency. Let conditions (2) and (9) - (12) be ful-

filled. Then relation (13) also holds for every x∈lλ

∞

and (τn)∈lµ

αby (9). Hence, Aλ∈(l∞, c), and the

finite limit ϕexists for every x∈lλ

∞by Lemma 3 ((a)

and (b)). Hence relation (14) holds for every x∈lλ

∞.

As (12) holds, then Aλ,µ ∈(l∞, lα)by Lemma 9.

Therefore, due to (9), A∈lλ

∞, lµ

α.

Remark 1. Conditions (10) and (11) can be replaced

by the condition

the series X

|ank|

λk

converges uniformly in n

(15)

in Theorem 1 by Lemma 3 ((a) and (c)).

Remark 2. If λn=O(1), then a matrix A= (ank)∈

lλ

∞, lµ

αfor α > 1if and only if conditions (2),

(9), (10) and (12) are satisfied. Indeed, in this case

(vk)∈c0for every x:= (xk)∈lλ

∞. Hence instead

of Aλ∈(l∞, c)we get Aλ∈(c0, c). Therefore in-

stead of Lemma 3 ((a) and (b)) we use now Lemma

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1. Moreover, instead of Aλ,µ ∈(l∞, c)in the present

case Aλ,µ ∈(c0, lα). As (c0, lα) = (l∞, c), then for

Aλ,µ we can use again Lemma 9 as we did in the proof

of Theorem 1.

Corollary 1. Condition (10) can be replaced by con-

dition

|ak|

λk

<∞(16)

in Theorem 1.

Proof. It is easy to see that condition (16) follows

from (2) and (10). In the same way, conditions (2),

(11) and (16) imply the validity of (10). Indeed, first

from condition (11) we obtain that

|ank −ak|

λk

=O(1).(17)

Since ank

λk

=ank −ak

λk

+ak

λk

then

|ank|

λk

≤X

|ank −ak|

λk

|ak|

λk

Moreover, the finite limits akexist by (2). Hence the

condition (10) is satisfied by (16) and (17).

Theorem 2. A matrix A= (ank)∈cλ

0, lµ

αfor α >

1if and only if conditions (2), (9), (10) and (12) are

satisfied.

Proof is similar to the proof of Theorem 1. The

only difference is that now Aλ∈(c0, c)and Aλ,µ ∈

(c0, lα). Therefore instead of Lemma 3 ((a) and (b))

we use Lemma 1 (for Aλ,µ ∈(c0, lα)we use again

Lemma 9 as in the proof of Theorem 1).

Theorem 3. A matrix A= (ank)∈lλ

1, lµ

αfor α >

1if and only if conditions (2), (9) are satisfied and

ank

λk

=O(1),(18)

λα

[µn|ank −ak|]α=O(1).(19)

Proof is similar to the proof of Theorem 1. The

only difference is that now Aλ∈(l1, c)and Aλ,µ ∈

(l1, lα). Therefore instead of Lemma 3 ((a) and (b))

we use Lemma 4, and instead of Lemma 9 we use

Lemma 8, considering that for Aλcondition (8) takes

the form (18), and for Aλ,µ ∈(l∞, lα)the condition

of Lemma 8 takes the form (19).

Corollary 2. Condition (18) can be replaced by con-

dition ak

λk

=O(1) (20)

in Theorem 3.

Proof is similar to the proof of Corollary 1, if to con-

sider that condition (20) follows from (2) and (18),

and condition (19) implies

ank −ak

λk

=O(1).

Theorem 4. A matrix A= (ank)∈cλ, lµ

αfor α >

1if and only if conditions (10), (12) are satisfied and

Ae ∈lµ

α, Aek∈lµ

α, Aλ−1∈lµ

α.(21)

Proof. Necessity. Suppose that A= (ank)∈

cλ, lµ

α. As ek∈cλ,e∈cλand λ−1∈cλ, then

condition (21) holds. As equality (13) holds for every

x:= (xk)∈cλ, and the finite limit

τ:= lim

nτn

exists due to Ae ∈lµ

α, then the matrix Aλtransforms

this convergent sequence (vk)into c. Similar to the

proof of the necessity of Theorem 1, it is possible to

show that, for every sequence (vk)∈c, there exists

a sequence (xk)∈cλsuch that vk=λk(xk−s).

Hence Aλ∈(c, c). This implies by Lemma 2 that the

finite limits akand

aλ:= lim

ank

λk

exist, and that condition (10) is satisfied. With the

help of (13), for every x∈cλ, we can write by Lemma

2 that

ϕ:= lim

nAnx=aλb+X

λk

(vk−b) + τs, (22)

where s:= limkxkand b:= limkvk. Now, using

(13) and (22), we obtain

µn(Anx−ϕ) = µnX

ank −ak

λk

(vk−b)

+µn(τn−τ)s+µn X

ank

λk

−aλ!b. (23)

As Ae ∈lµ

αand Aλ−1∈lµ

αby (10), then Aλ,µ ∈

(c0, lα). Therefore we can conclude by Lemma 9 that

condition (12) holds.

Sufficiency. Assume that conditions (10), (12) and

(21) are satisfied. First we notice that relation (13)

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holds for every x∈cλand the finite limits ak,τand

aλexist by (21). As (10) also holds, then Aλ∈(c, c)

by Lemma 2, and therefore relations (22) and (23)

hold for every x∈cλ. As condition (12) holds, then

Aλ,µ ∈(c0, lα)by Lemma 9. Moreover, Ae ∈lµ

αand

Aλ−1∈lµ

αby (21). Thus, A∈cλ, lµ

α.

Remark 3. Condition (12) can be replaced by the

condition

µα

n

X

k∈K∗

ank −ak

λk



<∞(24)

for arbitrary subset K∗of Nin Theorems 1, 2 and 4

by Lemma 9.

Example 1. Let us consider the Zweier matrix Z1/2,

defined by (ank), where (see [20], p. 14, or [1], p.

3) a00 = 1/2,ank = 1/2 if k=n−1or k=nfor

n≥1, and ank = 0 otherwise. The method A=Z1/2

is regular (see [1], p. 3). Let λbe defined by

λn:= (n+ 1)r, r > 0,(25)

and µby

µn:= (n+ 1)t, t > 0.(26)

Case 1: Z1/2 ∈lλ

∞, lµ

α∩cλ

0, lµ

α∩cλ, lµ

αfor

α > 1, if r < t −1/α. For proving it, we show that

all conditions of Theorems 1,2 and 4 hold. It is easy

to see that in this case ak= 0,τ= 1, and

Tn:= X

|ank|

λk

ank

λk

T0=1

2λ0

, Tn=1

21

λn−1

λn, n ≥1,(27)

then limnTn=aλ= 0, since λn=O(1).Therefore

conditions (2), (9) - (11) and (21) hold. Also condition

(12) is satisfied. Indeed,

S:= X

µα

n

X

k∈K

ank −ak

λk



≤µ0

2λ0

2α

∞

n=1

µα

n1

λn−1

λnα

for every possible Kfrom Nby (27). Hence, using

(25) and (26), we obtain

S=O(1)

∞

n=1

(n+ 1)rα 1

(n+ 1)tα

=O(1)

∞

n=1

(n+ 1)(t−r)α=O(1),

if (t−r)α > 1or r < t −1/α. Thus, condition (12)

holds.

Case 2: Z1/2 ∈lλ

1, lµ

αfor α > 1, if r≤t. For

proving it, we show that all conditions of Theorem 3

hold. The validity of (2) and (9) are proved in Case

1 of the present example; also it is easy to see that

condition (18) is satisfied. Let

V:= 1

λα

[µn|ank −ak|]α

2α

λα

kµα

k+µα

k+1.

Hence, using (25) and (26), we obtain

V=O(1) 1

(k+ 1)tα ((k+ 1)rα + (k+ 2)rα)

=O(1) 1

(k+ 1)(t−r)α=O(1),

if r≤t. Thus, condition (21) also holds.

In Example 3.1 rcan’t be greater than t; i.e.,

µn/λn=O(1). In the following example we con-

sider the case, where µn/λn=O(1) also is possible

for some collection of parameters.

Example 2. Let A= (ank )be a lower triangular

matrix defined by ank = 1/(n+ 1)c, c > 1, and λ,µ

respectively by (25) and (26).

Case 1: A∈lλ

∞, lµ

α∩cλ

0, lµ

α∩cλ, lµ

αfor

α > 1, if t > 1and r < c −1/α. For proving it, we

show that all conditions of Theorems 1,2 and 4 hold.

It is easy to see that in this case ak= 0,

τn=

k=0

ank =1

(n+ 1)c−1;τ= 0,

since c > 1. Thus conditions (2) and (9) hold. As

Tn=X

|ank|

λk

ank

λk

(n+ 1)c

k=0

(k+ 1)t,

then limnTn=aλ= 0 (since t > 1). Hence condi-

tions (10), (11) and (21) are satisfied. As

S=X

µα

n

X

k∈K

ank −ak

λk



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≤

∞

n=1

(n+ 1)rα "1

(n+ 1)c

k=0

(k+ 1)t#α

for every possible Kfrom N, then

S=O(1)

∞

n=1

(n+ 1)(c−r)α"n

k=0

(k+ 1)t#α

=O(1),

if (c−r)α > 1or r < c −1/α. Therefore condi-

tion (12) also holds. It is possible to find a collection

{α, c, r, t}with r > t satisfying conditions t > 1and

r < c −1/α. For example, if α= 2,c= 4 and

t= 2, then these conditions hold for r, satisfying the

relation 2< r < 3,5.

Case 2: A∈lλ

1, lµ

αfor α > 1, if r < c −1/α.

For proving it, we show that all conditions of Theorem

3 hold. The validity of (2) and (9) are proved in Case

1 of the present example; also it is easy to see that

condition (18) is satisfied. As

V=1

λα

[µn|ank −ak|]α

(k+ 1)tα

∞

n=k

(n+ 1)rα 1

(n+ 1)cα

(k+ 1)tα

∞

n=k

(n+ 1)(c−r)α,

then V=O(1), if (c−r)α > 1or r < c −1/α.

Therefore condition (21) also holds. There exists a

collection {α, c, r, t}with r > t satisfying the condi-

tion r < c −1/α. For example, if α= 2 and c= 4,

then for r, t, satisfying the relation 0< t < r < 3,5,

this condition holds.

3.2 The regularity of matrices on the sets

lλ

∞,cλ

0and cλ

We present necessary and sufficient conditions for the

regularity of a matrix Aon lλ

∞,cλ

0, and cλas the corol-

laries correspondingly from Theorems 1, 2 and 4.

Corollary 3. A matrix A= (ank)is regular on lλ

∞if

and only if condition (5) with τ= 1 is satisfied and

lim

|ank|

λk

= 0.(28)

Proof. Necessity. Assume that Ais regular on lλ

∞;

i.e., limnAnx=sfor every sequence x∈lλ

∞. Then

condition (5) with τ= 1 holds, since e∈lλ

∞, and

relation (13) holds for every x:= (xk)∈lλ

∞. This

implies that Aλtransforms every sequence (vk)∈l∞

into c0. Hence condition (28) is satisfied by Lemma

Sufficiency. Let all the conditions of the present

corollary are satisfied. Then relation (13) also holds

for every x∈lλ

∞. As condition (28) holds, then

Aλ∈(l∞, c0)by Lemma 5. Therefore limnAnx=s

for every x∈lλ

α, since τ= 1. Thus Ais regular on

lλ

∞.

Corollary 4. A matrix A= (ank )is regular on cλ

0if

and only if condition (2) with ak= 0,condition (5)

with τ= 1,and condition (10) are satisfied.

Proof is similar to the proof of Corollary 3, if to con-

sider that τ= 1 due to e∈cλ

0, and instead of Lemma

5 it is necessary to use Lemma 7, since in this case

Aλ∈(c0, c0).

Corollary 5. A matrix A= (ank)is regular on cλ

if and only if condition (2) with ak= 0,condition

(5) with τ= 1 and condition (10) are satisfied, and

aλ= 0.

Proof is similar to the proof of Corollary 3, if to con-

sider that τ= 1 due to e∈cλ, and instead of Lemma

5 it is necessary to use Lemma 6, since in this case

Aλ∈(c, c0).

Now we prove that there exists an irregular matrix,

which is regular on lλ

∞,cλ

0and cλ.

Example 3. Let A= (ank)be a matrix, where ank =

n+ 1 if k=n,ank =−nif k=n+ 1, and ank = 0

otherwise. Then obviously ak= 0 and τ= 1; i.e.,

conditions (2) and (5) hold. But condition (3) does

not hold, since

|ank|= 2n+ 1 =O(1) .

Thus the matrix Ais not regular by Lemma 2.

Case 1: Let λbe defined by (25) with r= 1.Then

Tn=X

|ank|

λk

= 1 + n

n+ 2 =O(1);

i.e., conditions (10) is satisfied. Hence Ais regular

on cλ

0by Corollary 4. Moreover,

ank

λk

= 1 −n

n+ 2,

aλ=lim

n1−n

n+ 2= 0.

Therefore Ais regular on cλby Corollary 5. We note

that Ais not regular on lλ

∞by Corollary 3, since con-

dition (11) does not hold.

Case 2: Let λbe defined by (25) with r= 2.Then

also condition (10) holds, aλ= 0, and in addition to

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.7

Ants Aasma, Pinnangudi N. Natarajan

E-ISSN: 2224-2880

Volume 23, 2024

it, limnTn= 0. Hence condition (11) is also satisfied

and in addition to regularity on cλ

0and on cλ,Ais also

regular on lλ

∞by Corollary 3.

4 Conclusion

In this paper we consider the α-absolute convergence

with speed, where the speed is defined by a monoton-

ically increasing positive sequence µand α > 1. The

notions of ordinary convergence and boundedness

with speed are known earlier. Let λbe another speed

of convergence, and lλ

∞,cλ

0,cλand lµ

αbe respectively

the sets of all λ-bounded, all λ-convergent to zero, all

λ-convergent and all α-absolutely µ-convergent se-

quences.

Let Abe a matrix with real or complex entries.

We found necessary and sufficient conditions for the

transforms A:lλ

∞→lµ

α,A:cλ

0→lµ

α,A:cλ→lµ

and A:lλ

1→lµ

αfor the case, when α > 1. As an ex-

ample we show that the Zweier matrix Z1/2 satisfies

these necessary and sufficient conditions for certain

speeds λand µ.

Also we define the notion of regularity on the sub-

space Xof the set of convergent sequences c, and

present necessary and sufficient conditions for a ma-

trix Ato be regular on lλ

∞,cλ

0and cλ. We presented

an example of irregular matrix, which is regular on cλ

and on cλ, but not on lλ

∞for some λ. Also we proved

that this irregular matrix is regular on cλ

0,cλand lλ

∞

for another λ.

Further we intend to define the notion of α-

absolute summability with speed by a matrix (with

real or complex entries), and to study the matrix trans-

forms between the sets of sequences, α-absolutely

summable with speeds by matrices.

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DOI: 10.37394/23206.2024.23.7

Ants Aasma, Pinnangudi N. Natarajan

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Volume 23, 2024

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Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Both authors equally contributed in this research, at

all stages from the formulation of the problem to the

final findings.

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scientific article itself

No funding was received for this study.

Conflicts of Interest

The authors have no conflicts of interest to declare.

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DOI: 10.37394/23206.2024.23.7

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E-ISSN: 2224-2880

Volume 23, 2024