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Article

Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method

1
Faculty of Materials Science and Technology in Trnava, Slovak University of Technology, Ulica Jána Bottu č. 2781/25, 917 01 Trnava, Slovakia
2
Faculty of Chemical and Food Technology in Bratislava, Slovak University of Technology, Radlinského 9, 812 37 Bratislava, Slovakia
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(3), 88; https://doi.org/10.3390/axioms11030088
Submission received: 4 January 2022 / Revised: 8 February 2022 / Accepted: 21 February 2022 / Published: 23 February 2022
(This article belongs to the Special Issue Approximation Theory and Related Applications)

Abstract

:
In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density d p ( A ) of the set A N is dependent on the sequence p = ( p n ) . Different sequences ( p n ) , for the same set A , will yield new and distinct densities. If the sequence ( p n ) does not differ from the sequence ( n ) in its order of magnitude, i.e., lim n p n n = 1 , then the resulting quasi-density is very close to the asymptotic density. The results for sequences that do not satisfy this condition are more interesting. In the next part, we deal with the necessary and sufficient conditions so that the quasi-statistical convergence will be equivalent to the matrix summability method for a special class of triangular matrices with real coefficients.

1. Introduction

The notion of asymptotic density for a subset of the set of natural numbers is known. It determines the size of the given subset compared to the set N .
Let A N . We define A ( n ) = | k A ,   k n | , i.e., as the number of elements of set A smaller than n .
Then
d _ ( A ) = lim inf n A ( n ) n
d ¯ ( A ) = lim sup n A ( n ) n
is the lower and upper asymptotic density of the set A N ,   respectively .
If d _ ( A ) = d ¯ ( A ) , then there exists lim n A ( n ) n = d ( A ) that is called the asymptotic density of set A . It is evident that if for some set A there exists d ( A ) , then 0 d ( A ) 1 (see [1]).
A different method for defining the density is based on the matrix method of limiting sequences of ones and zeros (see [2,3]).
Let
C = ( 1     0     0   0     0 1 2 1 2 0     0     0 1 n 1 n 1 n 1 n 0 )
be a regular (Cesáro) matrix (see [4]) defined as follows:
C = ( c n k ) ,
where
  c n k = 1 n               for   k n   c n k = 0                       for   k > n       n , k = 1 ,   2 ,  
Then, we define the asymptotic density of the set A N by the relation
d ( A ) = lim n k = 1 c n k · χ A ( k ) ,
where χ A ( k ) is the characteristic function of set A N (see [5,6]),
χ A ( k ) = { 1 ,     if   k A   0 ,     if   k A .
Agnew in [2] defined the sufficient condition for a matrix so that at least one sequence of ones and zeros to be limitable (summable) by the matrix.
Let A = ( a n k ) be an infinite matrix with real elements. The requirements of summability are:
(a)
k = 1 | a n k | M < ,   n = 1 ,   2 ,  
(b)
lim n max 1 k n | a n k | = 0 .
Let A = ( a n k ) be an infinite matrix with real elements. We say that the sequence x = ( x k ) is A limitable to the number s ϵ R ( A lim x k = s ), if lim n k = 1 a n k x k = s .
If the implication lim x k = s     A lim x k = s holds true, we say that the matrix A is regular [2].
The necessary and sufficient condition for the matrix A = ( a n k ) to be regular is
(a)
K > 0   n = 1 , 2 ,   k = 1 | a n k | K
(b)
k = 1 , 2 ,   lim n a n k = 0
(c)
lim n k = 1 a n k = 1 (see [6]).
Example 1.
If T = ( t n k ) is a regular matrix, then we can use it to define the density d T ( A .   ) (see [5,6]). Let the matrix T = ( t n k ) , where
      t n k = 1 k s n                                 for       k n       t n k = 0                                     for     k > n ,     n , k = 1 ,   2 ,  
and s n = j = 1 n 1 j , then
δ ( A ) = lim n k = 1 t n k · χ A ( k )
is the logarithmic density of the set A .
Let the matrix T = ( t n k ) , where
t n k = ϕ ( k ) n                                     for       k n ,   k n       t n k = 0                                               for     k n ,   k n         t n k = 0                                               for     k > n                             ,   n , k = 1 , 2 ,
and ϕ is the Euler function, then
d T ( A ) = lim n k = 1 t n k · χ A ( k )
is the Schoenberg density of the set A .
In this paper we define the quasi-density using a matrix, whose members satisfy special conditions.
In the next section, we will present the connection between statistical convergence and the matrix method of summability of the sequence of real numbers.
We say that x = ( x n ) converges statistically to the number L R , if
ε > 0 : d ( N ε ) = 0 ,   where   d ( N ε ) = { k ϵ N :   | x k L | ε } .
Numerous writers extended this convergence by substituting a different density for the asymptotic density (or by a function with suitable properties, respectively) (see [7,8,9,10,11]).
We will endeavor to characterize the quasi-statistical convergence by using the matrix method.
In the paper [12] the authors defined the quasi-statistical convergence as:
Let p = ( p n ) be a sequence of positive real numbers with the properties:
(a)
lim n p n = +
(b)
lim sup n p n n < + .
The quasi-density of the set A N for the sequence p = ( p n ) is
d p ( A ) = lim n 1 p n | { k ϵ A ,   k n   } | ,  
if such a limit exists.
If p n = n , then d p ( A ) is the asymptotic density of set A .
We say that the sequence x = ( x k ) converges quasi-statistically (given the sequence p = ( p n ) ) to the number L R ( stq p lim x k = L ), if ε > 0 the set E ε has a quasi-density equal to zero (t.j. d p ( E ε ) = 0 ), where E ε = { k ϵ N ,   | x k L | ε } .
If we define p n = n ,   n = 1 , 2 , , then the quasi-statistical convergence is identical to the statistical convergence.
If the sequence x = ( x n ) quasi-statistically converges to the number L , then it converges statistically as well. However, the reverse does not hold [12].
If lim inf n p n n > 0 , then if the sequence x = ( x n ) statistically converges to the number L , then it converges quasi-statistically as well (see [12]).

2. The Quasi-Density

Let p = ( p n ) be a sequence of positive real numbers that satisfies the following properties:
(a)
lim n p n = +
(b)
lim sup n p n n < + .
We will call such a sequence permissible.
The lower quasi-density of the set A N for a permissible sequence p = ( p n ) is
d p _ ( A ) = lim inf n 1 p n | { k ϵ A ,   k n } | ,  
if such a limit exists.
The upper quasi-density of the set A N for a permissible sequence p = ( p n ) is
d p ¯ ( A ) = lim sup n 1 p n | { k ϵ A ,   k n } | ,  
if such a limit exists.
In case d p _ ( A ) = d p ¯ ( A ) , then there exists a quasi-density of set A and we denote it as d p ( A ) = lim n 1 p n | { k ϵ A ,   k n } | .
Example 2.
Sequences that satisfy these properties (a permissible sequences) are, for example,
( p n ) = ( log n ) n = 1 ,     ( p n ) = ( n · α + d ) n = 1 ,   α R + ,   d R ,   ( p n ) = ( n α ) n = 1 ,   α ( 0 , 1 ) .
If the permissible sequence satisfies the following property, we can define the quasi-density of the set A using a matrix.
Let p = ( p n ) be a permissible sequence, let in addition lim inf n p n n = h ,   h R + ,   h 0 . We will create a matrix B = ( b n k ) as follows:
b n k = { 1 p n ,       k n 0 ,                 k > n ,   i . e . ,
B = ( 1 p 1     0     0   0     0 1 p 2 1 p 2 0     0     0 1 p n 1 p n 1 p n 1 p n 0 ) .
The matrix defined in this way does meet the Angew’s conditions.
It is true that
k = 1 b n k = 1 p n + 1 p n + + 1 p n = n · 1 p n = 1 p n n 1 h
and
lim n max 1 k n | b n k | = lim n 1 p n = 0 ,   because   lim n p n = .
Then, we can define the quasi-density of the set A N as follows:
Let χ A ( k ) be the characteristic function of set A .
Then,
d p _ ( A ) = lim inf n k = 1 b n k · χ A ( k )
and
d p ¯ ( A ) = lim sup n k = 1 b n k · χ A ( k )
are the lower and upper quasi-density of set A , respectively.
In case d p _ ( A ) = d p ¯ ( A ) , then there exists a quasi-density of set A and we denote it as d p ( A ) = d p _ ( A ) = d p ¯ ( A ) = lim n k = 1 b n k · χ A ( k ) .
We will now state several properties of a quasi-density.
Proposition 1.
Let A N be a finite set. Then, d p ( A ) = 0 for every permissible sequence p = ( p n ) .
Proof of Proposition 1.
If A is a finite set, then
d p ( A ) = lim n 1 p n | { k ϵ A ,   k n } | lim n | A | p n = 0 .
The quasi-density of the set of all natural numbers N is dependent on the sequence ( p n ) . □
Proposition 2.
Let ( p n ) be a permissible sequence.
(a)
If lim sup n p n n = T 0 , then d p ¯ ( N ) = 1 T (if lim sup n p n n = 1 , then d p ¯ ( N ) = 1 ).
(b)
If lim sup n p n n = 0 , then d p ¯ ( N ) = .
Proof of Proposition 2.
(a) Suppose that lim sup n p n n = T 0 . Then
d p ¯ ( N ) = lim sup n k = 1 b n k · χ k ( N ) = limsup n n p n = 1 T .
(b) Similarly
d p ¯ ( N ) = lim sup n k = 1 b n k · χ k ( N ) = limsup n n p n = limsup n 1 p n n = .  
Note: Let exists a finite lim n p n n
(a)
In the case of lim n p n n = L 0 , then d p ( N ) = 1 L .
(b)
In the case of lim n p n n = 0 , then d p ( N ) = .
(c)
In the case of lim n p n n = 1 , then d p ( N ) = 1 .
We see that, generally, for any A N : 0 d p _ ( A ) d p ¯ ( A ) + , i.e., quasi-density does not behave like any of the densities studied up to now.
If the sequence p = ( p n ) is such a permissible sequence, for which there exists a finite and non-zero limit lim n p n n , we can determine the relation between the asymptotic density and the quasi-density of a set.
Proposition 3.
Let A N be such a set, for which its asymptotic density is d ( A ) = m ,   m ϵ 0 , 1 . Let there exists a non-zero lim n p n n = L . Then, there also exists a quasi-density of set A and d p ( A ) = 1 L · m   holds true.
Proof of Proposition 3.
When we use the definition of quasi-density we get the following.
d p ( A ) = lim n 1 p n | { k ϵ A ,   k n } | = lim n n p n · 1 n | { k ϵ A ,   k n } | = 1 L · m .
Corollary 1.
Let p = ( p n ) be any arithmetic sequence of the type
p n = n · α + d ,   n = 1 , 2 , ,   α ϵ R + ,   d ϵ R .
Let A N be such a set that its asymptotic density d ( A ) = m ,   m ϵ R .
Then, d p ( A ) = m α .
Proof of Corollary 1.
For an arithmetic sequence the following applies:
lim n p n n = lim n n · α + d n = α .
From the previous theorem we obtain d p ( A ) = 1 α · m . □
Example 3.
Let A = { 1 2 ,   2 2 ,   , n 2 , } . It is evident that d ( A ) = 0 .
We define the sequence p = ( p n ) by:
p n = { 2 n ,                   n = 2 k n 2 ,           n = 2 k 1     k = 1 , 2 ,
This sequence satisfies the requirements of the definition. The quasi-density of set A = { 1 2 ,   2 2 ,   , n 2 , } given the sequence p is
d p ( A ) = lim n | A 1 , n | p n = lim n n p n = 0 .
Quasi-density of set B = { 1 , 2 , , n , } = N given the sequence p does not exist, because
d p _ ( B ) = lim inf n | N 1 , n | p n = lim inf n n p n = 2 ,
d p ¯ ( B ) = lim sup n | N 1 , n | p n = lim sup n n p n = 1 2 .
Example 4.
Let p n = log n ,   n = 2 , 3 , . It is evident that a sequence ( p n ) defined as such is permisible, because lim n log n = and lim n log n n = 0 .
Quasi-densities of the sets A = { 1 2 ,   2 2 ,   , n 2 , } and B = { 1 , 2 , , n , } given a sequence defined as preceding ( p n ) exists and is identical: d p ( A ) = a d p ( B ) = .
The asymptotic densities of these sets are not the same, because d ( A ) = 0 and d ( B ) = 1 . We can say the following corollary:
Corollary 2.
If lim sup n p n n = 0 ,then there is a set C N such that it exists d ( C ) ,but it does not exist d p ( C ) .
Proposition 4.
Let the following hold true for sequences p = ( p n )
0 < lim inf n p n n lim sup n p n n = T <
Then, for any sequence A N , 0 d p _ ( A ) d p ¯ ( A ) and d p ¯ ( A ) 1 T is valid.
Proof of Proposition 4.
0 d p _ ( A ) = lim inf n | k A , k n | p n = lim inf n n p n · | k A , k n | n
lim sup n n p n · | k A , k n | n = lim sup n | k A , k n | p n = d p ¯ ( A ) .
In addition to that d p ¯ ( A ) = lim sup n n p n · | k A , k n | n 1 T · d ¯ ( A ) 1 T . □
It is sufficient to realize that for every set A N there exists a d _ ( A ) and d ¯ ( A ) (an asymptotic density d ( A ) does not have to exist).
Corollary 3.
If there exists an asymptotic density d ( A ) of set A N , the the quasi-density d p ( A ) of this set exists if and only if lim n p n n = L 0 and d p ( A ) ϵ 0 , 1 L holds true.
In the next example, we will assume that lim n p n n = L 0 .
Proposition 5.
Let A , B N be a non-empty set for which their quasi-densities are d p ( A ) and d p ( B ) . Let d p : P N 0 , ) be a function. Then
(a)
If A B then d p ( A ) d p ( B ) .
(b)
d p _ ( A B ) d p ( A ) + d p ( B ) , d p ¯ ( A B ) d p ( A ) + d p ( B ) .
(c)
If A B = then d p ( A B ) = d p ( A ) + d p ( B ) .
Proof of Proposition 5.
(a) Let A B . Then, for every n N the following holds true
| { k ϵ A ,   k n } | | { k ϵ B ,   k n } | .
Then
1 p n | { k ϵ A ,   k n } | 1 p n | { k ϵ B ,   k n } | .
Transitioning to the limit, we obtain
lim n 1 p n | { k ϵ A ,   k n } |   lim n 1 p n | { k ϵ B ,   k n } | ,   i . e . ,   d p ( A ) d p ( B ) .
(b) It is evident that | { k ϵ A B ,   k n } | | { k ϵ A ,   k n } | + | { k ϵ B ,   k n } | .
From that we obtain the following:
d p _ ( A B ) = lim inf n 1 p n | { k ϵ ( A B ) ,   k n } | lim sup n 1 p n | { k ϵ ( A B ) ,   k n } |
lim sup n 1 p n ( | { k ϵ A ,   k n }     | + | { k ϵ B ,   k n }     | )
lim sup n 1 p n | { k ϵ A ,   k n } | + lim sup n 1 p n | { k ϵ B ,   k n } | = d p ¯ ( A ) + d p ¯ ( B ) =
= d p ( A ) + d p ( B ) .
(c) A B = | { k ϵ A B , k n } | = | { k ϵ A ,   k n } | + | { k ϵ B ,   k n } |
d p ¯ ( A B ) d p ( A ) + d p ( B ) = lim inf n 1 p n | { k ϵ A ,   k n } | + lim inf n 1 p n | { k ϵ B ,   k n } |
lim inf n 1 p n | { k ϵ ( A B ) ,   k n } | = d p _ ( A B ) .    
Now, we will show that quasi-densities have the almost Darboux property.
Definition 1.
We say that the density d ( A ) has the almost Darboux property, if for every real number t 0 , d ( N ) ) the exists such a set A N , for which its density is d ( A ) = t .
Theorem 1.
For every real number t 0 , ) there exists such a set A N and a permissible sequence p = ( p n ) , that d p ( A ) = t .
Proof of Theorem 1.
If t = 0 , then we can choose A to be any finite set (Proposition 1).
Let t ( 0 , ) , and let us choose any m ( 0 , 1 ] .
For these chosen immutable numbers, we define a sequence p = ( p n ) = ( m t · n ) n = 1 .
This sequence is permissible, because
lim n p n = lim n m t · n = +   a   lim sup n p n n = lim n m t · n n = m t < + .
An asymptotic density has the almost Darboux property:
For every m 0 , 1 ) exists such a set A N for which its asymptotic density is d ( A ) = m .
Let A be such a subset of natural numbers, such that its asymptotic density is m and the sequence ( p n ) = ( m t · n ) n = 1 . Then,
d p ( A ) = lim n 1 p n | { k ϵ A ,   k n } | = lim n t m · n | { k ϵ A ,   k n } | = t m lim n | { k ϵ A ,   k n     } | n = t m · m = t .
Theorem 2.
Let A N be such a subset of natural numbers, for which its asymptotic density d ( A ) = 0 . Let p = ( p n ) be a permissible sequence that satisfies the condition lim sup n p n n = T 0 . Then, d p ¯ ( A ) = 0 .
Proof of Theorem 2.
Let d ( A ) = 0 and lim sup n p n n = T 0 . The upper quasi-density of this set in regard to the sequence p = ( p n ) is
0 d p ¯ ( A ) = lim sup n | k A ,   k n | p n = lim sup n n p n · | k A ,   k n | n lim sup n n p n · lim sup n | k A ,   k n | n = 1 T · 0 = 0 .

3. The Quasi-Statistical Convergence and the Matrix Transformation

In the final part of this paper, we will focus on the quasi-statistical convergence of sequences of real numbers.
We will show the equivalence between this convergence and a matrix transformation of the same sequence.
Let p = ( p n ) be a permissible sequence. By T p we will denote the class of matrices with non-negative real members
B = ( b n k )       n , k = 1 , 2 ,
for which the following conditions are true:
(a)
k = 1 n b n k = 1
(b)
If D is a subset of natural numbers for which   d p ( D ) = 0 , then lim n k D b n k = 0 .
It is evident that if a matrix belong to the class T p , then it is regular. However, the reverse does not hold.
Example 5.
Let p = ( p n ) = ( 2 n + 1 ) n = 1 . Let the set D = { 1 2 , 2 2 , 3 2 , } . According to Proposition 3, the quasi-density of this set in regard to sequence p is
d p ( D ) = 1 L · m = 1 2 · 0 = 0 ,  
where L = lim n p n n = lim n 2 n + 1 n = 2 a m = d ( D ) = 0 .
Let us define the matrix C = ( c n k ) as follows:
c 11 = 1 ,   c 1 k = 0   for   k > 1                         c n k = 1 2 k log n   for   k D ,   k n c n k = 1 k log n   for   k D ,   k n         c n k = 0   for     k > n .
This matrix is the lower triangular regular, but do not belong to the class T p , because
k < n 2 k D b n k = 1 log n 2 ( 1 + 1 2 + + 1 n ) log n 2 log n = 1 2 0   for   n .
That is, if the matrix belongs to the class T p , so it is regular, the reverse is not true.
Lemma 1.
If the bounded sequence x = ( x k ) is not quasi-statistically convergent, then there exist real numbers λ < μ such that neither of the sets { n N ,   x n < λ } and { n N ,   x n > μ } has quasi-density zero.
Proof of Lemma 1.
The proof is the same as the proof of the Lemma in [6].
We will now utter a theorem that connects quasi-statistically convergent the sequences of real number and a matrix transformation of the same sequence using matrices from the class T p . □
Theorem 3.
The bounded sequence x = ( x k ) of real numbers is quasi-statistically convergent to L R ( st q lim x k = L ) if and only if it is summable to L R for each matrix B = ( b n k )   T p .
Proof of Theorem 3.
Let st q lim x k = L ,   L R and B = ( b n k )   T p .
As B is regular there exist a K R such that n = 1 , 2 ,   k = 1 | b n k | K .
It is sufficient to show that lim n a n = 0 where a n = k = 1 b n k · ( x k L ) .
For ε > 0 put B ( ε ) = { k N ,   | x k L | ε } .
By the assumption we have d p ( B ( ε ) ) = 0 we have lim k B ( ε ) k B ( ε ) | b n k | = 0 .
As the sequence x = ( x k ) is bounded, there exist M > 0 such that k = 1 , 2 , : | x k L | M .
Let ε > 0 . Then
| a n | k B ( ε 2 K ) | b n k | · | x k L | + k B ( ε 2 K ) | b n k | · | x k L |
M k B ( ε 2 K ) | b n k | + ε 2 K k B ( ε 2 K ) | b n k | M k B ( ε 2 K ) | b n k | + ε 2 .
By the condition (b) there exists an integer n 0 such that for all n > n 0 :  
k B ( ε 2 K ) | b n k | < ε 2 M .
Together we obtain lim n a n = 0 .
Conversely, suppose that st q lim x k = L does not apply. We show that it exists a matrix B = ( b n k )   T p such that B lim x k = L does not apply too.
If st q lim x k = L L then from the first part of proof it follows that B lim x k = L for any B T p . Thus we way assume that x = ( x k ) is not quasi-statistically convergent and by the above Lema there exist λ and μ ( λ < μ ) such that either the set U = { k N :   x k < λ } nor V = { k N :   x k > μ } has quasi-density zero.
It is clear that U V = . Let U n = U { 1 ,   2 ,   ,   n } and V n = V { 1 ,   2 ,   ,   n } . Therefore, there exists an ε > 0 and subsets U = { u k } k = 1 U and V = { v k } k = 1 V such for each k N : 1 u k · | { n N : n u k } | > ε and 1 v k · | { n N : n v k } | > ε .
Now define the matrix B = ( b n k ) in the following way
b n k = { 0                     k > n                                                     1 n                   k n     n U V 1 | U n |               k U n     n U                   1 | V n |               k V n     n V                  
Check that B = ( b n k ) T p . Obviously, B is a lower triangular nonnegative matrix. Condition (a) is clear from the definition of B , k = 1 n b n k = 1 for each n .
Condition (b) for this matrix:
Let for the set C N :   d p ( C ) = 0 .
Then,
k C n U V b n k = k C n U V 1 n χ k ( C ) = k C n U V p n n · 1 p n · χ k ( C ) .
For n we have
lim n k C n U V p n n · χ k ( C ) p n lim sup n p n n · d p ( C ) = 0
.
We proved that B = ( b n k ) belongs to T p .
Next, we will show B lim x k does not exist.
For n U :     k = 1 b n k · x k = k U n 1 | U n | · x k < λ · 1 = λ and for n V :   k = 1 b n k · x k = k V n 1 | V n | · x k > μ · 1 = μ . □

4. Conclusions

In this paper we define the lower quasi-density d p _ ( A ) , the upper quasi-density d p ¯ ( A ) and the quasi-density d p ( A ) of subsets of natural numbers, which we use to define the quasi-statistical convergence of sequences.
We proved some of the properties of quasi-densities (e.g., the quasi-density of a finite subset of natural numbers is zero and has the almost Darboux property). Given a permissible sequence, for which lim n p n n = L 0 there is a relation between the asymptotic and quasi-densities of set A , we have d p ( A ) = d ( A ) L .
The final section pertains to the quasi-statistical converge. We showed that the bounded sequence of real numbers is quasi-statistically convergent to L R if and only if it is summable to L R for each matrix B = ( b nk )   T p .
One of the most important applications of quasi-densities is connecting the quasi-statistical convergence with the summability method and by doing so generalize the term convergence.

Author Contributions

Conceptualization, R.M. and T.V.; formal analysis, R.M. and T.V.; methodology, R.M. and T.V.; validation, R.M. and T.V.; writing—original draft, R.M. and T.V.; writing—review and editing, R.M., T.V. and R.V. All authors have read and agreed to the published version of the manuscript.

Funding

This This article was written thanks to the generous support under the Operational Program Integrated Infrastructure for the project: “Strategic research in the field of SMART monitoring, treatment and preventive protection against coronavirus (SARS-CoV-2)”, Project no. 313011ASS8, co-financed by the European Regional Development Fund.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Paštéka, M. Density and Related Topics; Academia, Veda: Bratislava, Slovakia, 2017. [Google Scholar]
  2. Agnew, R.P. A simple sufficient condition that a method of summability be stronger than convergence. Bull. Am. Math. Soc. 1946, 52, 128–132. [Google Scholar] [CrossRef] [Green Version]
  3. Petersen, G.M. Regular Matrix Transformation; Mc Graw-Hill: London, UK, 1966. [Google Scholar]
  4. Freedman, A.R.; Sember, J.J. Densities and summability. Pac. J. Math. 1981, 95, 293–305. [Google Scholar] [CrossRef] [Green Version]
  5. Fridy, J.A. On statistical matrix convergence. Analysis 1985, 5, 301–313. [Google Scholar] [CrossRef]
  6. Fridy, J.A.; Miller, H.I. A matrix characterization of statistical convergence. Analysis 1991, 11, 59–66. [Google Scholar] [CrossRef]
  7. Kostyrko, P.; Šalát, T.; Wilczynski, W. I-convergence. Real Anal. Exch. 2000, 26, 669–686. [Google Scholar] [CrossRef]
  8. Mačaj, M.; Mišík, L.; Šalát, T.; Tomanová, J. On a class of densities of sets of positive integers. Acta Math. Univ. Comen. 2003, LXXII, 213–221. [Google Scholar]
  9. Baláž, V.; Šalát, T. Uniform density u and corresponding I u -convergence. Math. Commun. 2006, 11, 125–130. [Google Scholar]
  10. Gogola, M.; Mačaj, M.; Visnyai, T. On I ( q ) c -convergence. Ann. Math. Inform. 2011, 38, 27–36. [Google Scholar]
  11. Baláž, V.; Visnyai, T. I-convergence of aritmetical functions. In Number Theory and Its Applications; Intech Open Limited: London, UK, 2020; pp. 399–421. [Google Scholar]
  12. Ozguc, I.S.; Yurdakadim, T. On quasi-statistical convergence. Commun. Fac. Sci. Univ. Ank. Ser. A1 2012, 61, 11–17. [Google Scholar]
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Masarova, R.; Visnyai, T.; Vrabel, R. Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method. Axioms 2022, 11, 88. https://doi.org/10.3390/axioms11030088

AMA Style

Masarova R, Visnyai T, Vrabel R. Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method. Axioms. 2022; 11(3):88. https://doi.org/10.3390/axioms11030088

Chicago/Turabian Style

Masarova, Renata, Tomas Visnyai, and Robert Vrabel. 2022. "Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method" Axioms 11, no. 3: 88. https://doi.org/10.3390/axioms11030088

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