Lacunary arithmetic statistical convergence

Taja Yaying; Bipan Hazarika

arXiv:1703.03780·math.GM·March 13, 2017

Lacunary arithmetic statistical convergence

Taja Yaying, Bipan Hazarika

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TL;DR

This paper introduces new sequence spaces based on lacunary and arithmetic statistical convergence, studies their properties, and explores the concept of lacunary arithmetic statistical continuity.

Contribution

It defines the spaces ASC and ASC_θ, investigates their inclusion relations, and introduces lacunary arithmetic statistical continuity with key results.

Findings

01

Established inclusion properties between ASC and ASC_θ

02

Defined lacunary arithmetic statistical continuity

03

Proved several theorems on the properties of these spaces

Abstract

A lacunary sequence is an increasing integer sequence $θ = (k_{r})$ such that $k_{r} - k_{r - 1} \to \infty$ as $r \to \infty.$ In this article we introduce arithmetic statistically convergent sequence space $A S C$ and lacunary arithmetic statistically convergent sequence space $A S C_{θ}$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results.

Equations84

N_{θ} = {x = (x_{m}) \in w : r \to \infty lim \frac{1}{h _{r}} m \in I_{r} \sum ∣ x_{m} - l ∣ = 0 for some l} .

N_{θ} = {x = (x_{m}) \in w : r \to \infty lim \frac{1}{h _{r}} m \in I_{r} \sum ∣ x_{m} - l ∣ = 0 for some l} .

∥ x ∥_{N_{θ}} = r sup \frac{1}{h _{r}} m \in I_{r} \sum ∣ x_{m} ∣ .

∥ x ∥_{N_{θ}} = r sup \frac{1}{h _{r}} m \in I_{r} \sum ∣ x_{m} ∣ .

t \to \infty lim \frac{1}{t} ∣ {m \leq t : ∣ x_{m} - L ∣ \geq ε} ∣ = 0,

t \to \infty lim \frac{1}{t} ∣ {m \leq t : ∣ x_{m} - L ∣ \geq ε} ∣ = 0,

t \to \infty lim \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0.

t \to \infty lim \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0.

A S C = {(x_{m}) : t \to \infty lim \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0} .

A S C = {(x_{m}) : t \to \infty lim \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0} .

\frac{1}{t} {m \leq t : c x_{m} - c x_{⟨ m, n ⟩} \geq ε} = \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq \frac{ε}{∣ c ∣}}

\frac{1}{t} {m \leq t : c x_{m} - c x_{⟨ m, n ⟩} \geq ε} = \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq \frac{ε}{∣ c ∣}}

\frac{1}{t} {m \leq t : (x_{m} + y_{m}) - (x_{⟨ m, n ⟩} + y_{⟨ m, n ⟩}) \geq ε}

\frac{1}{t} {m \leq t : (x_{m} + y_{m}) - (x_{⟨ m, n ⟩} + y_{⟨ m, n ⟩}) \geq ε}

\leq \frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq \frac{ε}{2}} + \frac{1}{t} {m \leq t : y_{m} - y_{⟨ m, n ⟩} \geq \frac{ε}{2}} .

r \to \infty lim \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0.

r \to \infty lim \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0.

A S C_{θ} = {x = (x_{m}) : r \to \infty lim \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0} .

A S C_{θ} = {x = (x_{m}) : r \to \infty lim \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0} .

\frac{1}{h _{r}} {m \in I_{r} : c x_{m} - c x_{⟨ m, n ⟩} \geq ε} = \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq \frac{ε}{∣ c ∣}}

\frac{1}{h _{r}} {m \in I_{r} : c x_{m} - c x_{⟨ m, n ⟩} \geq ε} = \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq \frac{ε}{∣ c ∣}}

\frac{1}{h _{r}} {m \in I_{r} : (x_{m} + y_{m}) - (x_{⟨ m, n ⟩} + y_{⟨ m, n ⟩}) \geq ε}

\frac{1}{h _{r}} {m \in I_{r} : (x_{m} + y_{m}) - (x_{⟨ m, n ⟩} + y_{⟨ m, n ⟩}) \geq ε}

\leq \frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq \frac{ε}{2}} + \frac{1}{h _{r}} {m \in I_{r} : y_{m} - y_{⟨ m, n ⟩} \geq \frac{ε}{2}} .

k_{r - 1} < k_{r, 1}^{'} < k_{r, 2}^{'} < \dots < k_{η, η (r)}^{'} = k_{r}

k_{r - 1} < k_{r, 1}^{'} < k_{r, 2}^{'} < \dots < k_{η, η (r)}^{'} = k_{r}

lim I_{j}^{*} \subset I_{r} \sum \frac{1}{h _{j}^{*}} {m \in I_{j}^{*} : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0.

lim I_{j}^{*} \subset I_{r} \sum \frac{1}{h _{j}^{*}} {m \in I_{j}^{*} : x_{m} - x_{⟨ m, n ⟩} \geq ε} = 0.

\frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\frac{1}{h _{r}} {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\frac{∣ J _{j} ∣}{∣ I _{i} ∣} \geq δ for every J_{j} \subseteq I_{i} .

\frac{∣ J _{j} ∣}{∣ I _{i} ∣} \geq δ for every J_{j} \subseteq I_{i} .

(\frac{1}{∣ J _{j} ∣}) {m \in J_{j} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

(\frac{1}{∣ J _{j} ∣}) {m \in J_{j} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

= (\frac{∣ I _{i} ∣}{∣ J _{j} ∣}) (\frac{1}{∣ I _{i} ∣}) {m \in J_{j} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\leq (\frac{∣ I _{i} ∣}{∣ J _{j} ∣}) (\frac{1}{∣ I _{i} ∣}) {m \in I_{i} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\leq \frac{1}{δ} (\frac{1}{∣ I _{i} ∣}) {m \in I_{i} : x_{m} - x_{⟨ m, n ⟩} \geq ε},

\frac{∣ I _{ij} ∣}{∣ I _{i} ∣} \geq δ \mbox f or e v er y i, j = 1, 2, 3, \dots, I_{ij} \neq = ϕ .

\frac{∣ I _{ij} ∣}{∣ I _{i} ∣} \geq δ \mbox f or e v er y i, j = 1, 2, 3, \dots, I_{ij} \neq = ϕ .

\frac{h _{r}}{k _{r}} \geq \frac{σ}{1 + σ} .

\frac{h _{r}}{k _{r}} \geq \frac{σ}{1 + σ} .

\frac{1}{k _{r}} {m \leq k_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\frac{1}{k _{r}} {m \leq k_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\frac{τ _{r}}{h _{r}} < ε \mbox f or e v er y r \geq R .

\frac{τ _{r}}{h _{r}} < ε \mbox f or e v er y r \geq R .

\frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq ε}

\frac{1}{t} {m \leq t : x_{m} - x_{⟨ m, n ⟩} \geq ε}

1 < lim inf q_{r} \leq lim sup q_{r} < \infty,

1 < lim inf q_{r} \leq lim sup q_{r} < \infty,

A C_{θ} = {(x_{m}) : r \to \infty lim \frac{1}{h _{r}} m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩} = 0 for an integer n} .

A C_{θ} = {(x_{m}) : r \to \infty lim \frac{1}{h _{r}} m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩} = 0 for an integer n} .

m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩}

m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩}

\geq ∣ x_{m} - x_{⟨ m, n ⟩} ∣ \geq ε m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩} + ∣ x_{m} - x_{⟨ m, n ⟩} ∣ < ε m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩}

\geq ∣ x_{m} - x_{⟨ m, n ⟩} ∣ \geq ε m \in I_{r} \sum x_{m} - x_{⟨ m, n ⟩}

\geq ε {m \in I_{r} : x_{m} - x_{⟨ m, n ⟩} \geq ε}

r \to \infty lim \frac{1}{h _{r}} {m \in I_{r} : f_{N} (x_{m}) - f_{N} (x_{⟨ m, n ⟩}) \geq \frac{ε}{3}} = 0.

r \to \infty lim \frac{1}{h _{r}} {m \in I_{r} : f_{N} (x_{m}) - f_{N} (x_{⟨ m, n ⟩}) \geq \frac{ε}{3}} = 0.

{m \in I_{r} : f (x_{m}) - f (x_{⟨ m, n ⟩}) \geq \frac{ε}{3}}

{m \in I_{r} : f (x_{m}) - f (x_{⟨ m, n ⟩}) \geq \frac{ε}{3}}

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Full text

LACUNARY ARITHMETIC STATISTICAL CONVERGENCE

Taja Yaying1 and Bipan Hazarika*∗2*

1Department of Mathematics, Dera Natung Govt. College, Itanagar-791 111, Arunachal Pradesh, India

2Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791 112, Arunachal Pradesh, India

Email: [email protected]; [email protected]

Abstract.

A lacunary sequence is an increasing integer sequence $\theta=(k_{r})$ such that $k_{r}-k_{r-1}\rightarrow\infty$ as $r\rightarrow\infty.$ In this article we introduce arithmetic statistically convergent sequence space $ASC$ and lacunary arithmetic statistically convergent sequence space $ASC_{\theta}$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results.

Key Words: sequence space; lacunary; statistical convergence; arithmetic convergence.

AMS Subject Classification No (2010): 40A05, 40A99, 46A70, 46A99.

∗The corresponding author.

1. Introduction

A sequence $x=(x_{m})$ is called arithmetically convergent if for each $\varepsilon>0$ there is an integer $n$ such that for every integer $m$ we have $\left|x_{m}-x_{\left\langle m,n\right\rangle}\right|<\varepsilon,$ where the symbol $\left\langle m,n\right\rangle$ denotes the greatest common divisor of two integers $m$ and $n.$ We denote the sequence space of all arithmetic convergent sequence by $AC.$ The idea of arithmetic convergence was introduced by W.H.Ruckle [1]. The studies on arithmetic convergence and related results can be found in [1, 12, 13, 14, 15].

By a lacunary sequence we mean an increasing integer sequence $\theta=(k_{r})$ such that $h_{r}=k_{r}-k_{r-1}\rightarrow\infty$ as $r\rightarrow\infty$ . In this paper the intervals determined by $\theta$ will be denoted by $I_{r}=(k_{r-1},k_{r}]$ and also the ratio $\frac{k_{r}}{k_{r-1}},r\geq 1,k_{0}\neq 0$ will be denoted by $q_{r}$ . The space of lacunary convergent sequence $N_{\theta}$ was defined by Freedman [2] as follows:

[TABLE]

The space $N_{\theta}$ is a $BK$ -space with the norm

[TABLE]

The notion of lacunary convergence has been investigated by Çolak [5], Fridy and Orhan [3, 4], Tripathy and Et [6] and many others in the recent past.

The concept of statistical convergence was introduced by Fast[9] and later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy [7], Connor [8], Fridy and Orhan [4], Šalát [10] and many other authors.

A sequence $x=(x_{m})$ is said to be statistically convergent to the number $L$ if for every $\varepsilon>0,$

[TABLE]

where the vertical bars indicate the number of elements in the enclosed set.

The main purpose of this paper is to introduce arithmetic statistical convergence and lacunary arithmetic statistical convergence and to study some inclusion properties between these spaces. We also establish some sequential properties of lacunary arithmetic statistical continuity.

2. Main Results

A sequence $x=(x_{m})$ is said to be arithmetic statistically convergent if for $\varepsilon>0,$ there is an integer $n$ such that

[TABLE]

We shall use $ASC$ to denote the set of all arithmetic statistical convergent sequences. Thus for $\varepsilon>0$ and integer $n$

[TABLE]

We shall write $ASC-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ to denote the sequence $(x_{m})$ is arithmetic statistically convergent to $x_{\left\langle m,n\right\rangle}.$

Theorem 2.1.

Let $x=(x_{m})$ and $y=(y_{m})$ be two sequences.

(i).

If $ASC-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ and $c\in\mathbb{R},$ then $ASC-\lim cx_{m}=cx_{\left\langle m,n\right\rangle}.$ 2. (ii).

If $ASC-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ and $ASC-\lim y_{m}=y_{\left\langle m,n\right\rangle},$ then $ASC-\lim(x_{m}+y_{m})=(x_{\left\langle m,n\right\rangle}+y_{\left\langle m,n\right\rangle}).$

Proof.

(i). The result is obvious when $c=0$ . Suppose $c\neq 0,$ then for integer $n$

[TABLE]

which yields the result.

The result of (ii) follows from

[TABLE]

∎

Now we define a related concept of convergence in which the set $\left\{m:m\leq t\right\}$ is replaced by the set $\left\{m:k_{r-1}\leq m\leq k_{r}\right\},$ for some lacunary sequence $(k_{r}).$

Definition 1.

Let $\theta=(k_{r})$ be a lacunary sequence. The number sequence $x=(x_{m})$ is said to be lacunary arithmetic statistically convergent if for each $\varepsilon>0$ there is an integer $n$ such that

[TABLE]

We shall write

[TABLE]

We shall use $ASC_{\theta}-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ to denote the sequence $(x_{m})$ is lacunary arithmetic statistically convergent to $x_{\left\langle m,n\right\rangle}.$

Theorem 2.2.

Let $x=(x_{m})$ and $y=(y_{m})$ be two sequences.

(i).

If $ASC_{\theta}-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ and $c\in\mathbb{R},$ then $ASC_{\theta}-\lim cx_{m}=cx_{\left\langle m,n\right\rangle}.$ 2. (ii).

If $ASC_{\theta}-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ and $ASC_{\theta}-\lim y_{m}=y_{\left\langle m,n\right\rangle},$ then $ASC_{\theta}-\lim(x_{m}+y_{m})=(x_{\left\langle m,n\right\rangle}+y_{\left\langle m,n\right\rangle}).$

Proof.

(i). The result is obvious when $c=0$ . Suppose $c\neq 0,$ then for integer $n$

[TABLE]

which yields the result.

The result of (ii) follows from

[TABLE]

∎

Definition 2.

[2] Let $\theta=(k_{r})$ be a lacunary sequence. A lacunary refinement of $\theta$ is a lacunary sequence $\theta^{\prime}=(k^{\prime}_{r})$ satisfying $(k_{r})\subseteq(k^{\prime}_{r})$ .

Theorem 2.3.

If $\theta^{\prime}=(k^{\prime}_{r})$ is a lacunary refinement of a lacunary sequence $\theta=(k_{r})$ and $(x_{m})\in ASC_{\theta^{\prime}}$ then $(x_{m})\in ASC_{\theta}.$

Proof.

Suppose for each $I_{r}$ of $\theta$ contains the point $(k^{\prime}_{r,t})_{t=1}^{\eta(r)}$ of $\theta^{\prime}$ such that

[TABLE]

where $I^{\prime}_{r,t}=\left(k^{\prime}_{r,t-1},k^{\prime}_{r,t}\right].$

Since $(k_{r})\subseteq(k^{\prime}_{r}),$ , so $\forall r,\eta(r)\geq 1.$

Let $(I^{*})_{j=1}^{\infty}$ be the sequence of interval $(I^{*}_{r,t})$ ordered by increasing right end points. Since $(x_{m})\in ASC_{\theta^{\prime}},$ then for each $\varepsilon>0$ and an integer $n$

[TABLE]

Also since $h_{r}=k_{r}-k_{r-1},$ so $h^{\prime}_{r,t}=k^{\prime}_{r,t}-k^{\prime}_{r,t-1}.$

For each $\varepsilon>0$ and integer $n$

[TABLE]

This implies $(x_{m})\in ASC_{\theta}.$ ∎

Theorem 2.4.

Suppose $\beta=(l_{r})$ is a lacunary refinement of a lacunary sequence $\theta=(k_{r}).$ Let $I_{r}=\left(k_{r-1},k_{r}\right]$ and $J_{r}=\left(l_{r-1},l_{r}\right],r=1,2,\ldots$ If there exists $\delta>0$ such that

[TABLE]

Then $(x_{m})\in ASC_{\theta}\Rightarrow(x_{m})\in ASC_{\beta}.$

Proof.

For any $\varepsilon>0$ and integer $n$ and every $J_{j},$ we can find $I_{i}$ such that $J_{j}\subseteq I_{i},$ then we have

[TABLE]

which yields the result. ∎

Theorem 2.5.

Suppose $\beta=(l_{r})$ and $\theta=(k_{r})$ are two lacunary sequences. Let $I_{r}=\left(k_{r-1},k_{r}\right],~{}J_{r}=\left(l_{r-1},l_{r}\right],r=1,2,\ldots$ and $I_{ij}=I_{i}\cap J_{j},~{}i,j=1,2,3\ldots$ If there exists $\delta>0$ such that

[TABLE]

Then $(x_{m})\in ASC_{\theta}\Rightarrow(x_{m})\in ASC_{\beta}.$

Proof.

Let $\alpha=\beta\cup\theta.$ Then $\alpha$ is a lacunary refinement of $\theta.$ The interval sequence of $\alpha$ is $\left\{I_{ij}=I_{i}\cap I_{j}:I_{ij}\neq\phi\right\}.$ Using theorem 2.4 and the condition $\frac{\left|I_{ij}\right|}{\left|I_{i}\right|}\geq\delta$ yields that $(x_{m})\in ASC_{\theta}\Rightarrow(x_{m})\in ASC_{\alpha}.$ Since $\alpha$ is a lacunary refinement of the lacunary sequence $\beta,$ from theorem 2.3, we have $(x_{m})\in ASC_{\alpha}\Rightarrow(x_{m})\in ASC_{\beta}.$ ∎

There is a strong connection between the sequence space $ASC$ and $ASC_{\theta}.$ Now we give some inclusion relations between the spaces $ASC$ and $ASC_{\theta}.$

Theorem 2.6.

Let $\theta=(k_{r}),~{}r=1,2,3,\ldots$ be a lacunary sequence. If $\liminf q_{r}>1,$ then $ASC\subseteq ASC_{\theta}.$

Proof.

Let $(x_{m})\in ASC$ and $\liminf q_{r}>1.$ Then there exists $\sigma>0$ such that $q_{r}=\frac{k_{r}}{k_{r-1}}\geq 1+\sigma$ for sufficiently large $r,$ which implies that

[TABLE]

Then, for sufficiently large $r$ and integer $n,$

[TABLE]

Thus $x=(x_{m})\in ASC\Rightarrow(x_{m})\in ASC_{\theta}.$ ∎

Theorem 2.7.

For $\limsup q_{r}<\infty$ , we have $ASC_{\theta}\subseteq ASC.$

Proof.

Let $\limsup q_{r}<\infty$ then there exists $K>0$ such that $q_{r}<K$ for every $r.$ Let $\tau_{r}=\left|\left\{m\in I_{r}:\left|x_{m}-x_{\left\langle m,n\right\rangle}\right|\geq\varepsilon\right\}\right|$ where $n$ is an integer. Now for $\varepsilon>0$ and $(x_{m})\in ASC_{\theta}$ there exists $R$ such that

[TABLE]

Let $M=\max\left\{\tau_{r}:1\leq r\leq R\right\}$ and let $t$ be any integer with $k_{r}\geq t\geq k_{r-1}.$ Then for an integer $n$

[TABLE]

which yields $(x_{m})\in ASC.$ ∎

Corollary 2.8.

From Theorem 2.6 and Theorem 2.7, if $\theta=(k_{r})$ be a lacunary sequence and if

[TABLE]

then $ASC=ASC_{\theta}.$

In [11], Yaying and Hazarika introduced lacunary arithmetic convergent sequence space $AC_{\theta}$ as follows:

[TABLE]

We give some relation between the spaces $AC_{\theta}$ and $ASC_{\theta}.$

Theorem 2.9.

Let $\theta=(k_{r})$ be a lacunary sequence; then if $(x_{m})\in AC_{\theta}$ then $(x_{m})\in ASC_{\theta}.$

Proof.

Let $(x_{m})\in AC_{\theta}$ and $\varepsilon>0.$ We can write, for an integer $n$

[TABLE]

which gives the result. ∎

Corollary 2.10.

In view of [11, Theorem 2.2], if $(x_{m})\in AC$ then $(x_{m})\in ASC_{\theta}.$

3. Lacunary Arithmetic Statistical Continuity

In this section we shall introduce lacunary arithmetic statistical continuity and establish some interesting results.

Definition 3.

A function $f$ defined on a subset $E$ of $\mathbb{R}$ is said to be lacunary arithmetic statistical continuous if it preserves lacunary arithmetic statistical convergence i.e. if $ASC_{\theta}-\lim x_{m}=x_{\left\langle m,n\right\rangle}$ implies $ASC_{\theta}-\lim f(x_{m})=f(x_{\left\langle m,n\right\rangle}).$

We shall write $ASC_{\theta}$ continuous function to denote lacunary arithmetic statistical continuous function.

It is easy to see that the sum and the difference of two $ASC_{\theta}$ continuous functions is $ASC_{\theta}$ continuous. Also the composition of two $ASC_{\theta}$ continuous functions is again $ASC_{\theta}$ continuous. In the classical case, it is known that the uniform limit of sequentially continuous function is sequentially continuous, now we see that the uniform limit of $ASC_{\theta}$ continuous functions is also $ASC_{\theta}$ continuous.

Theorem 3.1.

Let $(f_{m})_{m\in\mathbb{N}}$ be a sequence of $ASC_{\theta}$ continuous functions defined on a subset $E$ of $\mathbb{R}$ and $(f_{m})$ be uniformly convergent to a function $f,$ then $f$ is $ASC_{\theta}$ continuous.

Proof.

Let $\varepsilon>0$ and $(x_{m})$ be any $ASC_{\theta}$ convergent sequence on a subset $E$ of $\mathbb{R}.$ By the uniform convergence of $f_{m},$ there exist $N\in\mathbb{N}$ such that $\left|f_{m}(x)-f(x)\right|<\frac{\varepsilon}{3}$ for all $m\geq N$ and for all $x\in E.$

Since $f_{N}$ is $ASC_{\theta}$ continuous on $E,$ we have for an integer $n$

[TABLE]

On the other hand, for an integer $n$ we have

[TABLE]

Thus it follows from the above inclusion that

[TABLE]

Thus $f$ is $ASC_{\theta}$ continuous. ∎

Theorem 3.2.

The set of all $ASC_{\theta}$ continuous functions defined on a subset $E$ of $\mathbb{R}$ is a closed subset of all continuous function on $E,$ i.e. $\overline{ASC_{\theta}(E)}$ = $ASC_{\theta}(E),$ where $ASC_{\theta}(E)$ denotes the set of all $ASC_{\theta}$ continuous functions defined on $E$ and $\overline{ASC_{\theta}(E)}$ denotes the closure of $ASC_{\theta}(E).$

Proof.

Let $f$ be any element of $\overline{ASC_{\theta}(E)}.$ Then there exists a sequence of points in $ASC_{\theta}(E)$ such that $\lim f_{m}=f.$ Now let $(x_{m})$ be any $ASC_{\theta}$ convergent sequence in $E.$ Since $(f_{m})$ converges to $f,$ there exists a positive integer $N$ such that

[TABLE]

Now $f_{N}$ is $ASC_{\theta}$ continuous on $E$ , so we have for an integer $n$

[TABLE]

On the other hand, for an integer $n$ we have

[TABLE]

From the above inclusion we can write

[TABLE]

Thus $f$ is $ASC_{\theta}$ continuous. So $f\in ASC_{\theta}(E)$ which gives us our required result. ∎

Bibliography15

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] W. H. Ruckle, Arithmetical Summability, Jour. Math. Anal. Appl. 396(2012) 741-748.
2[2] A. R. Freedman, J. J. Sember, M. Raphael, Some Cesàro-type summability spaces; Proc. Lond. Math. Soc. 37(1978) 508-520.
3[3] J. A. Fridy, C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl. 173(2)(1993) 497-504.
4[4] J. A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160(1)(1993) 43-51.
5[5] R. Çolak, Lacunary strong convergence of difference sequences with respect to a modulus function, Filomat 17(2003) 9-14.
6[6] B. C. Tripathy, M. Et, On generalized difference lacunary statistical convergence, Studia Univ. Babes-Bolyai Math. 50(1)(2005) 119-130.
7[7] J. A. Fridy, On statistical convergence. Analysis, 5(1985) 301-313.
8[8] J. Connor, The Statistical and strong p 𝑝 p -Cesaro convergence of sequences. Analysis, 8(1988) 47-63.

TL;DR

Contribution

Findings

Abstract

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Abstract.

1. Introduction

2. Main Results

Theorem 2.1**.**

Proof.

Definition 1**.**

Theorem 2.2**.**

Proof.

Definition 2**.**

Theorem 2.3**.**

Proof.

Theorem 2.4**.**

Proof.

Theorem 2.5**.**

Proof.

Theorem 2.6**.**

Proof.

Theorem 2.7**.**

Proof.

Corollary 2.8**.**

Theorem 2.9**.**

Proof.

Corollary 2.10**.**

3. Lacunary Arithmetic Statistical Continuity

Definition 3**.**

Theorem 3.1**.**

Proof.

Theorem 3.2**.**

Proof.

Theorem 2.1.

Definition 1.

Theorem 2.2.

Definition 2.

Theorem 2.3.

Theorem 2.4.

Theorem 2.5.

Theorem 2.6.

Theorem 2.7.

Corollary 2.8.

Theorem 2.9.

Corollary 2.10.

Definition 3.

Theorem 3.1.

Theorem 3.2.