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Restricted kinds of densities and associated ideals

Pratulananda Das

Department of Mathematics, Jadavpur University, West Bengal

Pratulananda Das Restricted kinds of densities and associated ideals

THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!!

Pratulananda Das Restricted kinds of densities and associated ideals

THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!!

• Let (X, ρ) be a metric space. Recall the definition of usual

convergence of a sequence (xn) in X.

Pratulananda Das Restricted kinds of densities and associated ideals

THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!!

• Let (X, ρ) be a metric space. Recall the definition of usual

convergence of a sequence (xn) in X. Definition 1. A sequence (xn) in X is said to converge to x0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ(xn, x0) < ε for all n ≥ m.

Pratulananda Das Restricted kinds of densities and associated ideals

THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!!

• Let (X, ρ) be a metric space. Recall the definition of usual

convergence of a sequence (xn) in X. Definition 1. A sequence (xn) in X is said to converge to x0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ(xn, x0) < ε for all n ≥ m.

• It is well known that a sequence is convergent if and only

if one of its tails converges to the same limit. Also for a sequence, the set of all limit points (subsequential limits) forms a closed set. • Consider the following sequences.

Pratulananda Das Restricted kinds of densities and associated ideals

THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!!

• Let (X, ρ) be a metric space. Recall the definition of usual

convergence of a sequence (xn) in X. Definition 1. A sequence (xn) in X is said to converge to x0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ(xn, x0) < ε for all n ≥ m.

• It is well known that a sequence is convergent if and only

if one of its tails converges to the same limit. Also for a sequence, the set of all limit points (subsequential limits) forms a closed set. • Consider the following sequences. Example 1. In a metric space (X, ρ) containing at least two points (for example R for simplicity) take two distinct points z and w. Let xn = z when n is even and xn = w when n is

• dd.

Pratulananda Das Restricted kinds of densities and associated ideals

THOUGH THIS IS A SET THEORY PAPER BUT I AM NOT A SET THEORIST. So let me describe how and why I was forced to take the help of SET THEORY !!!

• Let (X, ρ) be a metric space. Recall the definition of usual

convergence of a sequence (xn) in X. Definition 1. A sequence (xn) in X is said to converge to x0 ∈ X if for any ε > 0, there is a m ∈ N such that ρ(xn, x0) < ε for all n ≥ m.

• It is well known that a sequence is convergent if and only

if one of its tails converges to the same limit. Also for a sequence, the set of all limit points (subsequential limits) forms a closed set. • Consider the following sequences. Example 1. In a metric space (X, ρ) containing at least two points (for example R for simplicity) take two distinct points z and w. Let xn = z when n is even and xn = w when n is

• dd.

Example 2. Consider another sequence (yn) where yn = w if n = k2 for some k ∈ N and yn = z otherwise.

Pratulananda Das Restricted kinds of densities and associated ideals

• In order to understand the difference between the two

sequences, we will have to consider the following notion.

Pratulananda Das Restricted kinds of densities and associated ideals

• In order to understand the difference between the two

sequences, we will have to consider the following notion. Definition 2. By |A| we denote the cardinality of a set A. The lower and the upper natural densities of A ⊂ N are defined by d(A) = lim inf

n→∞

|A ∩ [1, n]| n and d(A) = lim sup

n→∞

|A ∩ [1, n]| n . If d(A) = d(A), we say that the natural density of A exists and it is denoted by d(A).

Pratulananda Das Restricted kinds of densities and associated ideals

• In order to understand the difference between the two

sequences, we will have to consider the following notion. Definition 2. By |A| we denote the cardinality of a set A. The lower and the upper natural densities of A ⊂ N are defined by d(A) = lim inf

n→∞

|A ∩ [1, n]| n and d(A) = lim sup

n→∞

|A ∩ [1, n]| n . If d(A) = d(A), we say that the natural density of A exists and it is denoted by d(A). Observation: We say that a subset of N is ”small” if it has natural density zero. Evidently any finite set has natural density zero. But now see that the set of odd integers as well as the set of even integers has density 1

2 whereas the

set of all squares has evidently density zero (just think that if A = {n : n = k2, k ∈ N} then within the the first n2 positive integers, A has n elements and so the density is equal to lim n

n2 .)

Pratulananda Das Restricted kinds of densities and associated ideals

• The following definition was introduced by Fast (1951),

Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers)

Pratulananda Das Restricted kinds of densities and associated ideals

• The following definition was introduced by Fast (1951),

Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Definition 3. A sequence (xn) in (X, ρ) is said to be statistically convergent to x0 ∈ X if for arbitrary ε > 0 the set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} has natural density zero.

Pratulananda Das Restricted kinds of densities and associated ideals

• The following definition was introduced by Fast (1951),

Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Definition 3. A sequence (xn) in (X, ρ) is said to be statistically convergent to x0 ∈ X if for arbitrary ε > 0 the set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} has natural density zero. Observation: Evidently it is now clear that the sequence (yn) defined above is statistically convergent to z (though is not convergent in the usual sense). Also it is easy to note that statistically convergent sequences need not be bounded as can be seen by observing the sequence (xn)

• f real numbers where xn = n if n = k2, k ∈ N and xn = 0,
• therwise.

Pratulananda Das Restricted kinds of densities and associated ideals

• The following definition was introduced by Fast (1951),

Steinhaus (1951) and before Zygmund (1936), Schoenberg (1959) all independently, with different names (for sequences of real numbers) Definition 3. A sequence (xn) in (X, ρ) is said to be statistically convergent to x0 ∈ X if for arbitrary ε > 0 the set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} has natural density zero. Observation: Evidently it is now clear that the sequence (yn) defined above is statistically convergent to z (though is not convergent in the usual sense). Also it is easy to note that statistically convergent sequences need not be bounded as can be seen by observing the sequence (xn)

• f real numbers where xn = n if n = k2, k ∈ N and xn = 0,
• therwise.
• Statistical convergence again resurfaced through the

major works by Salat (1980), Fridy (1885) and Connor (1989) and from then on the related investigations has become the most active research areas in Summability theory

Pratulananda Das Restricted kinds of densities and associated ideals

• We start by recalling the basic notions of ideals and filters.

Pratulananda Das Restricted kinds of densities and associated ideals

• We start by recalling the basic notions of ideals and filters.

Definition 4. A family I ⊂ 2Y of subsets of a non-empty set Y is said to be an ideal in Y if (i) A, B ∈ I implies A ∪ B ∈ I, (ii) A ∈ I, B ⊂ A imply B ∈ I.

Pratulananda Das Restricted kinds of densities and associated ideals

• We start by recalling the basic notions of ideals and filters.

Definition 4. A family I ⊂ 2Y of subsets of a non-empty set Y is said to be an ideal in Y if (i) A, B ∈ I implies A ∪ B ∈ I, (ii) A ∈ I, B ⊂ A imply B ∈ I. Observation: Further an admissible ideal I of Y satisfies {x} ∈ I for each x ∈ Y. Such ideals are also called free

• ideals. If I is a proper non-trivial ideal in Y (i.e.

Y / ∈ I, I = {φ}), then the family of sets F(I) = {M ⊂ Y : there exists A ∈ I : M = Y \ A} is a filter in Y. It is called the filter associated with the ideal I.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 5. A sequence (xn) is said to be I-convergent to x0 ∈ X (x0 = I − lim

n→∞xn) if and only if for each ε > 0 the

set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} ∈ I. The element x0 is called the I-limit of the sequence (xn)n. (Kostyrko, Salat, Wilczinsky, 2001)

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 5. A sequence (xn) is said to be I-convergent to x0 ∈ X (x0 = I − lim

n→∞xn) if and only if for each ε > 0 the

set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} ∈ I. The element x0 is called the I-limit of the sequence (xn)n. (Kostyrko, Salat, Wilczinsky, 2001)

• Recall the following result from the theory of statistical
• convergence. A sequence (xn) of real numbers is

statistically convergent to ξ if and only if there exist a set M = {m1 < m2 < ...} ⊂ N such that d(M) = 1 and lim

k→∞xmk = ξ (Salat, 1980).

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 5. A sequence (xn) is said to be I-convergent to x0 ∈ X (x0 = I − lim

n→∞xn) if and only if for each ε > 0 the

set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} ∈ I. The element x0 is called the I-limit of the sequence (xn)n. (Kostyrko, Salat, Wilczinsky, 2001)

• Recall the following result from the theory of statistical
• convergence. A sequence (xn) of real numbers is

statistically convergent to ξ if and only if there exist a set M = {m1 < m2 < ...} ⊂ N such that d(M) = 1 and lim

k→∞xmk = ξ (Salat, 1980).

• This result influenced the introduction of the following

concept of convergence, namely, I∗-convergence.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 5. A sequence (xn) is said to be I-convergent to x0 ∈ X (x0 = I − lim

n→∞xn) if and only if for each ε > 0 the

set K(ε) = {n ∈ N : ρ(xn, x0) ≥ ε} ∈ I. The element x0 is called the I-limit of the sequence (xn)n. (Kostyrko, Salat, Wilczinsky, 2001)

• Recall the following result from the theory of statistical
• convergence. A sequence (xn) of real numbers is

statistically convergent to ξ if and only if there exist a set M = {m1 < m2 < ...} ⊂ N such that d(M) = 1 and lim

k→∞xmk = ξ (Salat, 1980).

• This result influenced the introduction of the following

concept of convergence, namely, I∗-convergence. Definition 6. A sequence (xn) is said to be I∗-convergent to x0 ∈ X if and only if there exists a set M ∈ F(I), M = {m1 < m2 < ...} such that lim

k→∞ρ(xmk, x0) = 0. (Kostyrko, Salat, Wilczinsky, 2001)

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 7. An admissible ideal I is said to satisfy the condition (AP) (or is called a P-ideal or sometimes AP-ideal) if for every countable family of mutually disjoint sets (A1, A2, ...) from I there exists a countable family of sets (B1, B2, ...) such that Aj△Bj is finite for each j ∈ N and

∞

• k=1

Bk ∈ I.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 7. An admissible ideal I is said to satisfy the condition (AP) (or is called a P-ideal or sometimes AP-ideal) if for every countable family of mutually disjoint sets (A1, A2, ...) from I there exists a countable family of sets (B1, B2, ...) such that Aj△Bj is finite for each j ∈ N and

∞

• k=1

Bk ∈ I.

• It is clear that Bj ∈ I for each j ∈ N.

Theorem 3. Let (X, ρ) be a non-discrete metric space and I be an admissible ideal.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 7. An admissible ideal I is said to satisfy the condition (AP) (or is called a P-ideal or sometimes AP-ideal) if for every countable family of mutually disjoint sets (A1, A2, ...) from I there exists a countable family of sets (B1, B2, ...) such that Aj△Bj is finite for each j ∈ N and

∞

• k=1

Bk ∈ I.

• It is clear that Bj ∈ I for each j ∈ N.

Theorem 3. Let (X, ρ) be a non-discrete metric space and I be an admissible ideal. (i) If the ideal I has the property (AP) then for arbitrary sequence (xn) in X, I − lim xn = ξ implies I∗ − lim xn = ξ.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 7. An admissible ideal I is said to satisfy the condition (AP) (or is called a P-ideal or sometimes AP-ideal) if for every countable family of mutually disjoint sets (A1, A2, ...) from I there exists a countable family of sets (B1, B2, ...) such that Aj△Bj is finite for each j ∈ N and

∞

• k=1

Bk ∈ I.

• It is clear that Bj ∈ I for each j ∈ N.

Theorem 3. Let (X, ρ) be a non-discrete metric space and I be an admissible ideal. (i) If the ideal I has the property (AP) then for arbitrary sequence (xn) in X, I − lim xn = ξ implies I∗ − lim xn = ξ. (ii) If for every arbitrary sequence (xn) in X, I − lim xn = ξ implies I∗ − lim xn = ξ, then I has the property (AP).

Pratulananda Das Restricted kinds of densities and associated ideals

Theorem Let (X, ρ1) and (X, ρ2) be two metric spaces. Let I be a P−ideal which is not maximal. Then the following statements are equivalent. (i) The set of all ρ1 − I−convergent sequences coincides with the set of all ρ2 − I−convergent sequences. (ii) The set of all sequences convergent in (X, ρ1) coincides with the set of all sequences convergent in (X, ρ2). (iii) The metrics ρ1 and ρ2 induce one and the same topology

• n X.

Pratulananda Das Restricted kinds of densities and associated ideals

Let x = (xn)n∈N be a sequence of elements in a topological space (X, τ). y ∈ X is called an I-limit point of x if there exists a set M = {m1 < m2 < m3 < . . . } ⊂ N such that M / ∈ I and lim

k→∞xmk = y.

Pratulananda Das Restricted kinds of densities and associated ideals

Let x = (xn)n∈N be a sequence of elements in a topological space (X, τ). y ∈ X is called an I-limit point of x if there exists a set M = {m1 < m2 < m3 < . . . } ⊂ N such that M / ∈ I and lim

k→∞xmk = y.

Theorem (i) Let X be a first countable space. For any sequence (xn)n∈N in X the set I(Lx) is an Fσ-set provided I is an analytic P-ideal.

Pratulananda Das Restricted kinds of densities and associated ideals

Let x = (xn)n∈N be a sequence of elements in a topological space (X, τ). y ∈ X is called an I-limit point of x if there exists a set M = {m1 < m2 < m3 < . . . } ⊂ N such that M / ∈ I and lim

k→∞xmk = y.

Theorem (i) Let X be a first countable space. For any sequence (xn)n∈N in X the set I(Lx) is an Fσ-set provided I is an analytic P-ideal. (ii) Let X be a space with hcld(X) = ω. Then for each Fσ-set A in X there exists a sequence x = (xn)n∈N in X such that A = I(Lx) provided I is an analytic P-ideal.

Pratulananda Das Restricted kinds of densities and associated ideals

• The notion of natural density can be further extended as

follows.

Pratulananda Das Restricted kinds of densities and associated ideals

• The notion of natural density can be further extended as

follows.

• Let g : N → [0, ∞) be a function with lim

n→∞ g (n) = ∞. The

upper density of weight g was defined by the formula dg(A) = lim sup

n→∞

|A ∩ [1, n]| g (n) for A ⊂ N.

Pratulananda Das Restricted kinds of densities and associated ideals

• The notion of natural density can be further extended as

follows.

• Let g : N → [0, ∞) be a function with lim

n→∞ g (n) = ∞. The

upper density of weight g was defined by the formula dg(A) = lim sup

n→∞

|A ∩ [1, n]| g (n) for A ⊂ N.

• The family Ig = {A ⊂ N : dg(A) = 0} forms an ideal. It has

been observed that N ∈ Ig iff.

n g(n) → 0. So we additionally

assume that n/g (n) 0 so that N / ∈ Ig and it can be proved that Ig is a proper admissible P-ideal of N. The collection of all such functions g satisfying the above mentioned properties will be denoted by G.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 8. A submeasure on ω is a function ϕ: P(ω) → [0, ∞] such that:

ϕ(∅) = 0; if A ⊂ B then ϕ(A) ≤ ϕ(B), ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B), ϕ({n}) < ∞ for all n ∈ ω.

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 8. A submeasure on ω is a function ϕ: P(ω) → [0, ∞] such that:

ϕ(∅) = 0; if A ⊂ B then ϕ(A) ≤ ϕ(B), ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B), ϕ({n}) < ∞ for all n ∈ ω.

A submeasure ϕ is called a lower semicontinuos submeasure (in short, lscsm) if ϕ(A) = limn→∞ ϕ(A ∩ n) for all A ⊂ ω (this condition is equvalent to the classical lower semicontinuity of the function ϕ: 2ω → [0, ∞]).

Pratulananda Das Restricted kinds of densities and associated ideals

Definition 8. A submeasure on ω is a function ϕ: P(ω) → [0, ∞] such that:

ϕ(∅) = 0; if A ⊂ B then ϕ(A) ≤ ϕ(B), ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B), ϕ({n}) < ∞ for all n ∈ ω.

A submeasure ϕ is called a lower semicontinuos submeasure (in short, lscsm) if ϕ(A) = limn→∞ ϕ(A ∩ n) for all A ⊂ ω (this condition is equvalent to the classical lower semicontinuity of the function ϕ: 2ω → [0, ∞]). For any lscsm ϕ, we consider the exhaustive ideal given by Exh(ϕ) = {A ⊂ ω: lim

n→∞ ϕ(A \ n) = 0}.

It follows that for every lscsm ϕ on ω, Exh(ϕ) is an Fσδ P-ideal. A highly nontrivial theorem of Solecki states that each analytic P-ideal on ω is of the form Exh(ϕ) for some lscsm ϕ on ω.

Pratulananda Das Restricted kinds of densities and associated ideals

Theorem If g : ω → [0, ∞) is such that g(n) → ∞ and n/g(n) 0, then the ideal Zg is equal to Exh(ϕ) where ϕ(A) = sup

n∈ω

card(A ∩ n) g(n) for A ⊂ ω, and ϕ is a lower semicontinuos submeasure on ω. Consequently, Zg is an Fσδ P-ideal on ω.

Pratulananda Das Restricted kinds of densities and associated ideals

Theorem If g : ω → [0, ∞) is such that g(n) → ∞ and n/g(n) 0, then the ideal Zg is equal to Exh(ϕ) where ϕ(A) = sup

n∈ω

card(A ∩ n) g(n) for A ⊂ ω, and ϕ is a lower semicontinuos submeasure on ω. Consequently, Zg is an Fσδ P-ideal on ω. Theorem Let g1, g2 ∈ G be such that there exist M > 0 and k ∈ ω such that g1(n)/g2(n) ≤ M for all n ≥ k. Then Zg1 ⊂ Zg2. Consequently, if there exist 0 < m < M and k ∈ ω such that m ≤ g1(n)/g2(n) ≤ M for all n ≥ k, then Zg1 = Zg2.

Pratulananda Das Restricted kinds of densities and associated ideals

Theorem If g1, g2 ∈ G are such that n/g2(n) → ∞, g2(n)/g1(n) → ∞ then Zg1 Zg2. If g ∈ G and n/g(n) → ∞ then Zg Z.

Pratulananda Das Restricted kinds of densities and associated ideals

Theorem If g1, g2 ∈ G are such that n/g2(n) → ∞, g2(n)/g1(n) → ∞ then Zg1 Zg2. If g ∈ G and n/g(n) → ∞ then Zg Z.

• Let 0 < α < β ≤ 1 and g1(n) = nα, g2(n) = nβ for n ∈ ω.

Then Zg1 Zg2.

Pratulananda Das Restricted kinds of densities and associated ideals

Theorem If g1, g2 ∈ G are such that n/g2(n) → ∞, g2(n)/g1(n) → ∞ then Zg1 Zg2. If g ∈ G and n/g(n) → ∞ then Zg Z.

• Let 0 < α < β ≤ 1 and g1(n) = nα, g2(n) = nβ for n ∈ ω.

Then Zg1 Zg2.

• There exists a function g ∈ G such that Zg Z and Zg is

different from any ideal generated by a function of the form nα with 0 < α < 1. Let ak := 22k and g(n) :=

• (ak)

1 2

for n ∈ [ak, ak+1) and k = 1, 2, . . . 1 for n < 4.

• Clearly, g ∈ G and n/g(n) → ∞ which implies that Zg Z.

We can show that Zg is incomparable with Zn1/3.

Pratulananda Das Restricted kinds of densities and associated ideals

Define g(n) := (2k)! for (2k − 1)! < n ≤ (2k)! and k = 1, 2, . . . n for (2k)! < n ≤ (2k + 1)! and k = 1, 2, . . . . Since g(n) → ∞ and g((2k)!)/(2k)! = 1 for all k, we see that g ∈ G. Also n/g(n) ≤ 1 for all n, hence we have Z ⊂ Zg. We can show that this inclusion is proper.

Pratulananda Das Restricted kinds of densities and associated ideals

Define g(n) := (2k)! for (2k − 1)! < n ≤ (2k)! and k = 1, 2, . . . n for (2k)! < n ≤ (2k + 1)! and k = 1, 2, . . . . Since g(n) → ∞ and g((2k)!)/(2k)! = 1 for all k, we see that g ∈ G. Also n/g(n) ≤ 1 for all n, hence we have Z ⊂ Zg. We can show that this inclusion is proper. Define f(n) := (2k)! for (2k − 1)! < n ≤ (2k + 1)! and k = 1, 2, . . . . Observe that f ∈ G. We will show that Zf is incomparable with Z.

Pratulananda Das Restricted kinds of densities and associated ideals

Define g(n) := (2k)! for (2k − 1)! < n ≤ (2k)! and k = 1, 2, . . . n for (2k)! < n ≤ (2k + 1)! and k = 1, 2, . . . . Since g(n) → ∞ and g((2k)!)/(2k)! = 1 for all k, we see that g ∈ G. Also n/g(n) ≤ 1 for all n, hence we have Z ⊂ Zg. We can show that this inclusion is proper. Define f(n) := (2k)! for (2k − 1)! < n ≤ (2k + 1)! and k = 1, 2, . . . . Observe that f ∈ G. We will show that Zf is incomparable with Z. Theorem There exists a family G0 ⊂ G of cardinality c such that Zf is incomparable with Z for every f ∈ G, and Zf and Zg are incomparable for any distinct f, g ∈ G0.

Pratulananda Das Restricted kinds of densities and associated ideals

[1] M. Balcerzak, P . Das, M. Filipczak, J. Swaczyna, Generalized kinds of density and the associated ideals, Acta

• Math. Hungar, 147 (1) (2015), 97 - 115.

[2] S. Bhunia, P . Das and S. K. Pal, Restricting statistical convergence, Acta Math. Hungar., 134 (1-2) (2012), 153-161. [3] H. Cartan, Th´ eorie des filtres, C. R. Acad. Sci. Paris, 205 (1937), 595–598. [4] Pratulananda Das, Some further results on ideal convergence in topological spaces, Topology Appl., 159 (10-11) (2012), 2621 - 2626. [5] Pratulananda Das, E. Savas, On Generalized Statistical and Ideal Convergence of Metric valued Sequences, Ukrainian

• Math. J., 68 (12) (2017), 1849 - 1859.

[6] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.

Pratulananda Das Restricted kinds of densities and associated ideals

[7] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313. [8] P . Kostyrko, T. ˇ Sal´ at, W. Wilczy´ nski, I-convergence, Real

• Anal. Exchange, 26 (2000-2001), 669–686.

[9] M. K¨ uc¨ ukaslan, U. Deger, O. Dovgoshey, On the statistical convergence of metric valued sequences, Ukrainian Math. J., 66 (5) (2014), 712-720. [10] B. K. Lahiri and Pratulananda Das, I-convergence and I∗-convergence in topological spaces, Math. Bohem., 130 (2) (2005), 153–160. [11] T. ˇ Salˇ at, On statistically convergent sequences of real numbers, Math. Slovaca, 30, (1980), 139–150. [12] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66, (1959), 361–375. [13] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74.

Pratulananda Das Restricted kinds of densities and associated ideals