# AsymptoticI-Equivalence of Two Number Sequences and Asymptotic I-Regular Matrices

Authors: Hafize Gumus; Jeff Connor; Fatih Nuray
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

We studyI-equivalence of the two nonnegative sequences x
=
x
k and y
=
y
k. Also we define asymptotic I-regular matrices and obtain conditions for a matrix A
=
a
j
k to be asymptotic I-regular.


---

## Body

## 1. Introduction

The notion ofI-convergence was introduced by Kostyrko et al. for real sequences (see [1]) and then extended to metric spaces by Kostyrko et al. (see [2]). Fast [3] introduced statistical convergence and I-convergence, which is based on using ideals of ℕ to define sets of density 0; is a natural extension of Fast’s definition.Definition 1.
A family of setsI
⊆
2
ℕ is called an ideal if and only if(i)
∅
∈
I;
(ii)
for eachA
,
B
∈
I we have A
∪
B
∈
I;
(iii)
for eachA
∈
I and each B
⊆
A we have B
∈
I.
An ideal is called nontrivial ifℕ
∉
I and a nontrivial ideal is called admissible if {
n
}
∈
I for each n
∈
ℕ (see [2]).Definition 2.
A family of setsF
⊂
2
ℕ is a filter in ℕ if and only if(i)
∅
∉
F;
(ii)
for eachA
,
B
∈
F we have A
∩
B
∈
F;
(iii)
for eachA
∈
F and each B
⊇
A we have B
∈
F (see [2]).Proposition 3.
I is a nontrivial ideal in ℕ if and only if
(1)
F
=
F
(
I
)
=
{
M
=
ℕ
∖
A
:
A
∈
I
}
is a filter inℕ (see [2]).Definition 4.
A real sequencex
=
(
x
k
) is said to be I-convergent to L
∈
ℝ if and only if for each ε
>
0 the set
(2)
A
ε
=
{
k
∈
ℕ
:
|
x
k
-
L
|
≥
ε
}
belongs toI. The number L is called the I-limit of the sequence x (see [2]).Letx
=
(
x
k
) and y
=
(
y
k
) be real sequences. Pobyvanets introduced asymptotic equivalence for x and y as follows: if
(3)
lim
⁡
k
→
∞
⁡
x
k
y
k
=
1
,thenx and y are called asymptotic equivalent; this is denoted by x
~
y (see [4]). Pobyvanets also introduced asymptotic regular matrices which preserve the asymptotic equivalence of two nonnegative number sequences; that is, for the nonnegative matrix A
=
(
a
j
k
) if x
~
y then A
x
~
A
y (see [5]).Theorem 5.
LetA
=
(
a
j
k
) be a nonnegative matrix. A is asymptotic regular if and only if, for each m,
(4)
lim
⁡
j
→
∞
⁡
a
j
m
∑
k
=
1
∞
a
j
k
=
1
(see [5]).The frequency of terms having zero values makes a term-by-term ratiox
k
/
y
k inapplicable in many cases, which motivated Fridy to introduce some related notions. In particular, he analyzed the asymptotic rates of convergence of the tails and partial sums of series as well as the supremum of the tails of bounded sequences (see [6]).Definel
1, P
0, P, and P
δ spaces as follows:
(5)
l
1
=
{
x
=
(
x
k
)
:
∑
k
=
1
∞
‍
|
x
k
|
<
∞
}
.
P
δ
=
{
The
set
of
all
real
number
sequences
such
that
iiiiiiiii
x
k
≥
δ
>
0
∀
k
}
,
P
0
=
{
the
set
of
all
nonnegative
sequences
which
have
iiiiiiii
at
most
a
finite
number
of
zero
terms
}
,
P
=
{
the
set
of
all
sequences
x
such
that
x
k
>
0
∀
k
}
.Following Fridy, given a sequencex
=
(
x
k
), the sequence R
x is defined to be (
R
m
x
) where, for each m
∈
ℕ, R
m
x
=
∑
k
=
m
∞
x
k.Theorem 6.
IfA is a nonnegative (
l
∞
,
l
1
) summability matrix, then the following statements are equivalent:(i)
ifx and y are bounded sequences such that x
~
y and y
∈
P
δ, for some δ
>
0, then R
A
x
~
R
A
y;
(ii)
for eachm,
(6)
lim
⁡
n
→
∞
⁡
(
∑
k
=
n
∞
a
k
m
∑
k
=
n
∞
∑
j
=
0
∞
a
k
j
)
=
0(see [7]).In 2003, Patterson extended these concepts by introducing asymptotically statistical equivalent sequences, an analog of the above definitions, and investigated natural regularity conditions for nonnegative summability matrices (see [4]).Definition 7.
The sequencex has statistical limit L, denoted by st-lim
⁡
x
=
L provided that for every ε
>
0,
(7)
lim
⁡
n
⁡
1
n
{
the
number
of
k
≤
n
:
|
x
k
-
L
|
≥
ε
}
=
0
(see [4]).Definition 8.
Two nonnegative sequencesx and y are said to be asymptotically statistically equivalent provided that, for every ε
>
0,
(8)
lim
⁡
n
⁡
1
n
{
the
number
of
k
≤
n
:
|
x
k
y
k
-
1
|
≥
ε
}
=
0
.
In this case we writex
~
S
y (see [4]).Definition 9.
A summability matrixA is asymptotically statistically regular provided that A
x
~
S
A
y whenever x
~
S
y, x
∈
P
0, and y
∈
P
δ for some δ
>
0 (see [4]).Having introduced these ideas, Patterson then offered characterizations of(
i
) in Theorem 6 when a nonnegative summability matrix A maps bounded sequences into the absolutely convergent sequences and has the property that if x
∈
P
0, y
∈
P
δ, and x
~
S
y, then R
A
x
~
S
R
A
y and (
ii
) when a summability matrix is asymptotically statistical regular summability matrices.Theorem 10.
In order for a summability matrixA to be asymptotically statistical regular it is necessary and sufficient that(i)
∑
k
=
1
k
0
a
n
k is bounded for each n;
(ii)
for any fixedk
0 and ε
>
0,
(9)
lim
⁡
n
→
∞
1
n
{
t
h
e
n
u
m
b
e
r
o
f
k
≤
n
:
|
∑
k
=
1
k
0
a
n
k
∑
k
=
1
∞
a
n
k
|
≥
ε
}
=
0(see [4]).The main results of this paper have a similar focus, where statistical convergence is replaced by convergence with respect to an admissible ideal of subsets ofℕ.


## 2. Main Results

Definition 11.
Letx and y be nonnegative real sequences and let I be an admissible ideal in ℕ. If
(10)
{
k
∈
ℕ
:
|
x
k
y
k
-
1
|
≥
ε
}
belongs toI, for every ε
>
0, then x and y are called asymptotically I-equivalent sequences; this is denoted by x
~
I
y.Theorem 12.
LetA
=
(
a
j
k
) be a nonnegative (
l
∞
,
l
1
) summability matrix and let I and J be admissible ideals in ℕ. Then the following statements are equivalent:(i)
ifx and y are bounded sequences such that x
~
I
y, x
∈
P
0, and y
∈
P
δ, for some δ
>
0, then R
A
x
~
J
R
A
y;
(ii)
J
-
lim
⁡
n
(
(
∑
j
≥
n
∑
k
∈
S
a
j
k
)
/
(
∑
j
≥
n
∑
k
=
1
∞
a
j
k
)
)
=
0 for each S
∈
I.Proof.
The proof that statement(
ii
) implies statement (
i
) is given first. Assuming that x
~
I
y, let ε
>
0 be given and set
(11)
S
=
{
k
:
|
x
k
y
k
-
1
|
≥
ε
2
}
.
Observe thatS
∈
I and that
(12)
(
1
-
ε
2
)
y
k
<
x
k
<
(
1
+
ε
2
)
y
k
for eachk
∉
S. By (
ii
),(13)
T
=
{
n
∈
ℕ
:
|
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
|
<
ε
2
δ
sup
⁡
k
(
x
k
)
}
∈
F
(
J
)
.
Observe that, forn
∈
T,
(14)
R
n
(
A
x
)
R
n
(
A
y
)
=
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
x
k
)
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
y
k
)
=
∑
j
≥
n
(
∑
k
∈
S
a
j
k
x
k
+
∑
k
∉
S
a
j
k
x
k
)
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
y
k
)
≤
sup
⁡
k
(
x
k
)
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
y
k
)
+
(
1
+
(
ε
/
2
)
)
∑
j
≥
n
∑
k
∉
S
a
j
k
y
k
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
y
k
)
≤
sup
⁡
k
(
x
k
)
δ
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
+
(
1
+
ε
2
)
<
sup
⁡
k
(
x
k
)
δ
(
ε
2
δ
sup
⁡
k
(
x
k
)
)
+
(
1
+
ε
2
)
=
1
+
ε
.
That is,(15)
R
n
(
A
x
)
R
n
(
A
y
)
≤
1
+
ε
.
On the other hand,(16)
R
n
(
A
x
)
R
n
(
A
y
)
=
∑
j
≥
n
(
∑
k
∈
S
a
j
k
x
k
+
∑
k
∉
S
a
j
k
x
k
)
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
y
k
)
≥
δ
sup
⁡
k
(
y
k
)
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
+
(
1
-
(
ε
/
2
)
)
∑
j
≥
n
∑
k
∉
S
a
j
k
x
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
y
k
≥
δ
sup
⁡
k
(
y
k
)
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
+
(
1
-
(
ε
/
2
)
)
∑
j
≥
n
∑
k
=
1
∞
a
j
k
y
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
y
k
-
(
1
-
(
ε
/
2
)
)
∑
j
≥
n
∑
k
∈
S
∞
a
j
k
y
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
y
k
≥
δ
sup
⁡
k
(
y
k
)
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
+
(
1
-
ε
2
)
-
(
1
-
(
ε
/
2
)
)
(
sup
⁡
k
(
y
k
)
)
∑
j
≥
n
∑
k
∈
S
∞
a
j
k
δ
∑
j
≥
n
∑
k
=
1
∞
a
j
k
.
By condition(
ii
), there is a set U
∈
F
(
J
) such that, for each n
∈
U, the first and third terms of the above expression can be made small in relation to 1
-
(
ε
/
2
) and, in particular,
(17)
R
n
(
A
x
)
R
n
(
A
y
)
≥
1
-
ε
for eachn
∈
U. By (15) and (17), we have
(18)
J
-
lim
⁡
⁡
R
n
(
A
x
)
R
n
(
A
y
)
=
1
and hence(
ii
) implies (
i
).
The proof is completed by showing that statement(
i
) implies statement (
ii
). Assume that if x and y are bounded sequences, x
∈
P
0, y
∈
P
δ, and x
~
I
y, then R
A
x
~
J
R
A
y. Let S be a member of I and define the sequences x and y as follows:
(19)
x
k
=
{
1
,
if
k
∉
S
,
0
,
if
k
∈
S
,
y
k
=
1
.
Observe that(20)
R
n
(
A
x
)
R
n
(
A
y
)
=
∑
j
≥
n
(
∑
k
∈
S
a
j
k
x
k
+
∑
k
∉
S
a
j
k
x
k
)
∑
j
≥
n
(
∑
k
=
1
∞
a
j
k
y
k
)
=
∑
j
≥
n
∑
k
=
1
∞
a
j
k
-
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
=
1
-
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
and, asR
A
x
~
J
R
A
y, it follows that
(21)
J
-
lim
⁡
⁡
∑
j
≥
n
∑
k
∈
S
a
j
k
∑
j
≥
n
∑
k
=
1
∞
a
j
k
=
0
.Definition 13.
LetA
=
(
a
j
k
) be a nonnegative summability matrix and let I be an admissible ideal in ℕ. Further assume that x and y are nonnegative real sequences with x
∈
P and y
∈
P
δ for some δ
>
0. The summability matrix A is said to be asymptotic I-regular if x
~
I
y implies A
x
~
I
A
y.Lemma 14.
Letx
=
(
x
k
) and y
=
(
y
k
) belong to P
δ
∩
l
∞ and let I be an admissible ideal in ℕ. Then necessary and sufficient condition for x
~
I
y is I
-
lim
⁡
⁡
(
x
-
y
)
=
0.Proof.
Assuming thatx
~
I
y, then
(22)
T
=
{
k
:
|
x
k
y
k
-
1
|
<
ε
}
∈
F
(
I
)
and, for eachk
∈
T, we have(23)
(
1
-
ε
)
y
k
<
x
k
<
(
1
+
ε
)
y
k
and hence(24)
(
1
-
ε
)
y
k
≤
x
k
≤
(
1
+
ε
)
y
k
⟹
-
ε
y
k
<
x
k
-
y
k
<
ε
y
k
⟹
-
ε
(
sup
⁡
k
|
y
k
|
)
<
x
k
-
y
k
<
ε
(
sup
⁡
k
|
y
k
|
)
⟹
-
ε
′
<
x
k
-
y
k
<
ε
′
for
each
k
∈
T
.
HenceI
-
lim
⁡
k
(
x
k
-
y
k
)
=
0.Now suppose thatI
-
lim
⁡
k
(
x
k
-
y
k
)
=
0. Then(25)
B
ε
=
{
k
:
|
x
k
-
y
k
|
<
ε
}
∈
F
(
I
)
.For eachk
∈
B
ε, we have
(26)
-
ε
<
x
k
-
y
k
<
ε
⟹
-
ε
y
k
<
x
k
y
k
-
1
<
ε
y
k
⟹
-
ε
δ
<
x
k
y
k
-
1
<
ε
δ
⟹
1
-
ε
δ
<
x
k
y
k
<
1
+
ε
δ
⟹
{
k
:
|
x
k
y
k
-
1
|
<
ε
δ
}
∈
F
(
I
)and hencex
~
I
y.Definition 15.
LetI be an admissible ideal in ℕ, x a real sequence, and A
=
(
a
j
k
) a nonnegative summability matrix. If I
-
lim
⁡
⁡
x
=
L implies I
-
lim
⁡
⁡
(
A
x
)
=
L, then A is called an I-regular matrix.Theorem 16.
LetA
=
(
a
j
k
) be a nonnegative summability matrix, A
∈
(
l
∞
,
l
∞
), and I an admissible ideal. The matrix A is I-regular if and only if(i)
{
j
:
|
∑
k
∈
S
a
j
k
|
≥
ε
}
∈
I for all S
∈
I;
(ii)
I
-
lim
⁡
∑
k
=
1
∞
a
j
k
=
1.The proof of Theorem 16 is similar to [3, Theorem 2.1].Theorem 17.
LetI
⊆
2
ℕ be an admissible ideal in ℕ. Then a nonnegative summability matrix A
=
(
a
j
k
) is asymptotic I-regular if and only if
(27)
I
-
lim
⁡
∑
k
∈
S
a
j
k
∑
k
=
1
∞
a
j
k
=
0
for everyS
∈
I.Proof.
Suppose thatA is asymptotic I-regular. Let S
∈
I and define the sequences x and y as follows:(28)
x
k
=
{
1
,
if
k
∉
S
,
0
,
if
k
∈
S
,
y
k
=
1
∀
k
∈
ℕ
.
Observe thatx and y are bounded sequences, x
∈
P
0, and y
∈
P
δ. Hence. 							(29)
(
A
x
)
j
(
A
y
)
j
=
∑
k
∈
S
a
j
k
x
k
+
∑
k
∉
S
a
j
k
x
k
∑
k
=
1
∞
a
j
k
y
k
=
∑
k
=
1
∞
a
j
k
-
∑
k
∈
S
a
j
k
∑
k
=
1
∞
a
j
k
=
1
-
∑
k
∈
S
a
j
k
∑
k
=
1
∞
a
j
k
.
SinceA
x
~
I
A
y, it follows that(30)
I
-
lim
⁡
∑
k
∈
S
a
j
k
∑
k
=
1
∞
a
j
k
=
0
.
Next we show that condition (27) is sufficient for A to be asymptotic I-regular. Define the matrix B
=
(
b
j
k
) by b
j
k
=
a
j
k
/
∑
k
=
1
∞
a
j
k. Note that, since the row sums equal 1 and condition (27) yields criteria (
i
) of Theorem 16, B is an I-regular matrix. Also observe that, since the row sums equal 1, the matrix B maps members of P
δ to P
δ. Observe that
(31)
x
~
I
y
⟹
I
-
lim
⁡
⁡
(
x
-
y
)
=
0
⟹
I
-
lim
⁡
⁡
B
(
x
-
y
)
=
0
⟹
I
-
lim
⁡
⁡
(
B
x
-
B
y
)
=
0
⟹
B
x
~
I
B
y
and hence(32)
(
B
x
)
j
(
B
y
)
j
=
∑
k
=
1
∞
b
j
k
x
k
∑
k
=
1
∞
b
j
k
y
k
=
∑
k
=
1
∞
(
a
j
k
/
∑
k
=
1
∞
a
j
k
)
x
k
∑
k
=
1
∞
(
a
j
k
/
∑
k
=
1
∞
a
j
k
)
y
k
=
∑
k
=
1
∞
a
j
k
x
k
∑
k
=
1
∞
a
j
k
y
k
.
SinceB
x
~
I
B
y, it follows that A
x
~
I
A
y.



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*Source: 805857-2014-02-24.xml*